tag:blogger.com,1999:blog-57187363650001602232024-03-13T02:19:43.871-07:00Statistics et al.Statistical education, publishing, sports analytics, and game theory - everything that makes math useful in real life. Now carbon negative!Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.comBlogger176125tag:blogger.com,1999:blog-5718736365000160223.post-67939827181138608362023-12-29T17:20:00.000-08:002023-12-29T17:20:20.361-08:00Rating Systems Explained<p>Teams in league sports often play balanced schedules. For example, in each season of the English Premier League, each football/soccer team plays each other team in the league exactly twice - once at their home stadium, and once at the opposing stadium. That way, at the end of the season, you can merely look at the record of wins, draws, and losses to determine which teams have done the best, second best, and so on. This works because everyone has had the same opposition, so wins against that same opposition are comparable.<br /><br />But what would happen if different teams played different opponents, or even different numbers of matches? This is exactly the situation in many eSports, as well as chess; individual players may play different amounts of time against completely different opponents. To compare competitors in such as situation, we can use a rating system.<br /><br /> <span></span></p><a name='more'></a><br /><br />A typical rating system assigns a score to each competitor, which is either an individual or a team, depending on the game being played. Higher scores indicate better long term performance by that competitor. When two competitors face each other, points are lost by the loser and gained by the winner, but the amount won or lost depends on how 'surprising' the result is.<br /><br /><br />For example, if a strong competitor (pre-match rating 1400) beats a weak competitor (pre-match rating 1000), perhaps only 2 points of rating would be exchanged (updated ratings of 1402 and 998, respectively), because this result was estimated to be the most likely based on previous performance. However, if the weak competitor wins, many more points of rating would be exchanged, say 16 points, updating the ratings to 1384 for the strong competitor and 1016 for the weak competitor. A draw can be considered half a win for each competitor, so a draw between unequal opponents would also result in an exchange of points; in our example the strong competitor would lose 7 (the average of gaining 2 and losing 16) points to the weak competitor (making the new ratings 1393 and 1007, respectively.).<br /><br /><br />In the long run, we would expect the strong competitor to win much more often, and thus gain a smaller number of points more often than the weak competitor. As such, the ratings should stabilize at their true relative values for each competitor after many matches.<br /><br /><br />The most famous, of these ratings systems is the Elo system, developed by Arpad Elo. Players across the world have ratings, and these ratings are used in part to determine tournament eligibility (e.g., a tournament may have an "Under 1600" division and an open division), or determine titles like grandmaster.<br /><br /><br />Elo has an implementation in R through the "elo" package on CRAN. There is documentation and some vignettes, including an introduction at <a href="https://cran.r-project.org/web/packages/elo/index.html">https://cran.r-project.org/web/packages/elo/index.html</a> . The "Calculating Running Elo Updates" has a classic example of two-team matches as well as an example for four-team competition.<br /><br /><br /><br /><br />Glicko (details at: <a href="http://www.glicko.net/glicko/glicko.pdf">http://www.glicko.net/glicko/glicko.pdf</a> ), developed by Mark Glickman, assigns competitors both a rating and an uncertainty measure. As the a competitor plays in more matches, their uncertainty measure is reduced because we have more information about their true rating. As time passes, that information becomes less relevant, so the uncertainty measure increases. Ratings change more with each match when uncertainty is high.<br /><br /><br />Glicko, Elo, and other rating systems are implemented in the 'PlayerRatings' package on CRAN, with details here <a href="https://cran.r-project.org/web/packages/PlayerRatings/">https://cran.r-project.org/web/packages/PlayerRatings/</a> .<br /><br /><br /><br /><br />Other rating systems incorporate other details. RAPTOR (details at: <a href="https://fivethirtyeight.com/features/introducing-raptor-our-new-metric-for-the-modern-nba/">https://fivethirtyeight.com/features/introducing-raptor-our-new-metric-for-the-modern-nba/</a> ), developed by Nate Silver of 538, used player tracking data in NBA basketball to estimate offense and defense ability of individual players; RAPTOR then uses individual players in the aggregate to give a rating to a team. <p></p><p> </p><p>Glicko 2.0 incorporates winning or losing streaks, taking them as predictive indications that a competitor is actually better/worse than their wins/losses suggest individually.<br /><br /><br />Microsoft Trueskill 1 and 2 are particularly interesting because they manage competitions between more than two competitors, such as a multiplayer video game. Trueskill 1 uses only the ranking of players; to oversimplify, a player wins against everyone they rank higher than in a multiplayer match and lose against everyone they rank lower than. Trueskill 2 uses more detailed player performances like in-game score to determine adjustments to ratings. (Details of Trueskill 1 are at: <a href="https://www.microsoft.com/en-us/research/wp-content/uploads/2007/01/NIPS2006_0688.pdf">https://www.microsoft.com/en-us/research/wp-content/uploads/2007/01/NIPS2006_0688.pdf</a> , and Trueskill 2 are at: <a href="https://www.microsoft.com/en-us/research/uploads/prod/2018/03/trueskill2.pdf">https://www.microsoft.com/en-us/research/uploads/prod/2018/03/trueskill2.pdf</a> )<br /><br /><br />The 'TrueSkillThroughTime' package on CRAN implements the TrueSkill rating system and expands upon it. (See <a href="https://github.com/glandfried/TrueSkillThroughTime.R">https://github.com/glandfried/TrueSkillThroughTime.R</a> for details and an example). The simpler package 'trueskill' has been removed from CRAN because it was last updated in 2013.<br /><br /><br />These ratings systems are used to for predictions. In the FIDE implementation of Elo, a rating difference of 400 suggests a 10-to-1 advantage, and a 200-point difference suggests a square root of 10, or 3.16-to-1 advantage. That is, a player rated 1200 is expected to get 76% of the wins against a 1000-rated player, and a 24% of the wins against a 1400-rated player. In terms of American odds, if there is a difference of 200 points, fair odds would be -316 for the higher rated player, and +316 for the lower rated player, ignoring the chance of a draw. Likewise, in the classical implementation of Glicko, a difference of 400 points suggests a 10-to-1 advantage.<br /><br /><br />The predictions from Trueskill 2 (and Trueskill 1 in games older than 2018) are used for matchmaking purposes. When a player enters a queue for a match, they are paired with other players such that the estimated probability of that player overperforming or underperforming compared to opposing and teammate players is low.<br /><br /><br /><b><i>In part 2 of this article, found in my gambling course to be released soon on Udemy, we will apply Glicko 1 to NHL ice hockey teams, and a system like Trueskill to horse races at Woodbine racetrack in Toronto.</i></b><br /><br /><br />See also: <a href="https://towardsdatascience.com/developing-a-generalized-elo-rating-system-for-multiplayer-games-b9b495e87802">https://towardsdatascience.com/developing-a-generalized-elo-rating-system-for-multiplayer-games-b9b495e87802</a> (Developing a generalized Elo Rating system for 3+ player games)<br /><br /><br />and <a href="https://uwaterloo.ca/computational-mathematics/sites/ca.computational-mathematics/files/uploads/files/justin_dastous_research_paper.pdf">https://uwaterloo.ca/computational-mathematics/sites/ca.computational-mathematics/files/uploads/files/justin_dastous_research_paper.pdf</a> (Trueskill with in-game events)<br /><br /><br />and <a href="https://cs.stanford.edu/people/paulliu//files/www-2021-elor.pdf">https://cs.stanford.edu/people/paulliu//files/www-2021-elor.pdf</a> (Elo for massive multiplayer)<br /><br /></p>Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-67797911100432315252023-12-26T09:11:00.000-08:002023-12-26T09:11:00.499-08:00Reflection on an undergrad/grad crosslisted design of experiments course<p> </p><p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<span style="font-size: small;">I'm looking to re-examine and
improve my teaching since I started working as a professor again for
the University of Waterloo, and part of that means writing
reflections on courses again. This one is for a 4<sup>th</sup> year /
graduate level crosslisted course on design and analysis of
experiments. </span>
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p><span><a name='more'></a></span>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<span style="font-size: large;">What worked:</span></p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Asking for feedback around week 4-5.</b> Students felt heard.
They had some concerns that I was able to address. Namely, that I was
teaching in a data science style when I should have been focusing
more on mathematical rigour.</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
Also, I was able to pivot my lecturing style to from typing in ms
paint to writing on paper, which seemed to work. Over the holidays, I
bought an artist's tablet and I'm learning to use it effectively, so
soon I'll be able to eliminate the paper and just upload handwritten
annotations directly.</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Digital textbook.</b> Montgomery is the go-to book for this
course, and thankfully an older version of the book was available
digitally through the library. I managed to rely on a textbook while
saving the class a lot of money.</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Recording class.</b> This continues to be well received.
Although I need to find a recording system that works without the
internet. Currently I'm using Zoom.</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<span style="font-size: large;">What didn't:</span></p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Leaving semester prep to the last minute.</b> This is especially
true because Design of Experiments has always challenged me -
factorial designs aren't something I use in practice much, so I was
reliant on the text book and Nathaniel Steven's notes a lot. In my defense, I was taking care of an 8-month old without any day care
available yet, but that challenge will always be there in some form
and I need to learn to work around it.</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Pacing.</b> I ran out of planned material with 3 lectures to go,
so we did a lot of final exam review. This would have been better if
it was spread throughout the semester as additional examples.</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Treating 430 and 830 as a the same class. </b>I wonder if there
are any resources for crosslisted classes. I wonder if I should write
one. Would make a good M.Ed. thesis topic.
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<span style="font-size: large;">Challenges (excuses and
whining):</span></p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- 1.5 new preps this term. (Stat 230 isn't really a new prep because
it's so familiar)</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- Baby</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- Constantly sick</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<span style="font-size: large;">Challenges (actionable)</span></p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Lack of deep knowledge about Stat 430</b></p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Balancing exam difficulty.</b> It's hard to come up with
questions that some people will get and others won't in a way that's
fair. This comes back to the lack of expertise. I think I skewed on
the easy side by looking at the results. Waterloo students are VERY
smart as well, so an exam that would work elsewhere tends to skew
easy in these classes. It's also not a huge problem in a 4xx/8xx
class to have a 78% midterm average because the 8xx students have
substantially higher grade requirements anyways.</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Classroom engagement.</b> This might be an effect of the student
body involved, but I had very little direct engagement during class.
I had quite a bit of student activity on piazza, and I must remember to be more
communicative on there - it's not enough just to address the problems
people ask on piazza, they need to see the response right there on
piazza that something was done.</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
<br />
</p>
<p align="left" class="western" style="line-height: 100%; margin-bottom: 0cm;">
- <b>Participating in Marking.</b> Still on the fence about this.
This class has a lot of written answers on tests and assignments, and
I've taken to marking a lot of those myself instead of relying on
TAs. It takes time, some of which is offset by reduced remark
requests. It also gives TAs less experience with marking written
questions, and I should remember that I'm also in the business of
training them.</p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-56624792369821785372022-03-08T16:28:00.003-08:002022-03-09T12:40:40.826-08:00 Reading Assignment – Collecting Carefully.<span style="font-size: 12.0pt; line-height: 107%;">This is a reading assignment for </span><span style="font-size: 12.0pt; line-height: 107%;"><span style="font-size: 12.0pt; line-height: 107%;">“Episode 28: Collect Carefully” of
“The Data Science Ethics Podcast”, available at: </span><a href="https://datascienceethics.com/podcast/collect-carefully/"><span style="font-size: 12.0pt; line-height: 107%;">https://datascienceethics.com/podcast/collect-carefully/</span></a><span style="font-size: 12.0pt; line-height: 107%;"> </span>. My eventual hope is to incorporate it into a course on data ethics and (AI) safety, but that's still a long way from being anything solid. From recent interviews with post secondary institutions, I heard that a lot of schools are looking to incorporate ethics into their stats and data science courses, so I hope this and some of my future posts can contribute to those efforts.<br /></span><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> <span></span></span></p><a name='more'></a><p></p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">For this
reading assignment, read the transcript to “Episode 28: Collect Carefully” of
“The Data Science Ethics Podcast”, available at: </span><a href="https://datascienceethics.com/podcast/collect-carefully/"><span style="font-size: 12.0pt; line-height: 107%;">https://datascienceethics.com/podcast/collect-carefully/</span></a><span style="font-size: 12.0pt; line-height: 107%;"> and answer the following questions.
Answers can be verbatim from the transcript unless they say ‘in your own words’
or are follow-up questions.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">There are 8
comprehension-level questions and 3 deeper follow-up questions that require
deeper investigation or thought.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Note: Make
sure to click on the “View full episode transcript” tab inside the page.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Note: The
transcription on the webpage is good, but imperfect. (e.g., “forcing that feel
to be completed” should read “forcing that FIELD to be completed”.) If
something doesn’t make sense, the audio download of the podcast episode is
available on the webpage too. It’s 15-20 minutes at normal speed.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q1. In your
own words, what does “pre-hoc” mean?</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q2. What are
two reasons that you don’t include data/variables just for the sake having
them? (i.e., including them without have a good reason)</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q3. What is
an example of a variable that could be used as a proxy for race?</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q3F (follow-up).
What are two other variables not mentioned in the transcript that could be used
as a proxy for a protected class? (e.g., sex, gender, ethnicity, religion,
sexual orientation, age above 45)</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q4. What is
a common problem with AI and machine learning algorithms that were trained on
visual data of humans?</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q4F
(follow-up). Describe in your own words the “Google Gorillas” problem being
alluded to in this transcript? (There another episode transcript about the
problem at </span><a href="http://datascienceethics.com/podcast/google-gorilla-problem-photo-tagging-algorithm-bias/"><span style="font-size: 12.0pt; line-height: 107%;">http://datascienceethics.com/podcast/google-gorilla-problem-photo-tagging-algorithm-bias/</span></a><span style="font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">
</span>, but there are many other sources about this)</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q5. What an
example of an “adversary” problem that can come from retail worker incentives
to collect data, such as emails?</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q6. What
does Marie imply by using the term “clean data”?</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q6F
(follow-up). Compare this definition of “clean data” to the tidyverse definition
of “tidy data”. (How are these two definitions different, despite sounding so
similar)</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q7. What are
two ways that the design of your survey can affect the answers that you
collect?</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Q8. What are
some ways that sensor data can be become biased?</span></p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">(Message me at mj2davis@uwaterloo.ca for an answer key) <br /></span></p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-91724271659227876802022-02-17T15:57:00.007-08:002022-02-17T15:57:50.949-08:00Peer review of "Algorithmically deconstructing shot locations as a method for shot quality in hockey"<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; line-height: 107%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">This is an open peer review I did for the manuscript "</span><span style="font-family: "Times New Roman",serif;">Algorithmically deconstructing shot
locations as a method for shot quality in hockey</span><span style="font-family: "Times New Roman",serif; font-size: 12.0pt; line-height: 107%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">" by </span><span style="font-family: "Times New Roman",serif;">Devan G. Becker, Douglas G.
Woolford and Charmaine B. Dean as submitted to the Journal of Quantitative
Assessment in Sport</span><span style="font-family: "Times New Roman",serif; font-size: 12.0pt; line-height: 107%; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">, back in 2020.</span><span style="font-family: "Times New Roman",serif;"></span>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">You can
find the manuscript behind a $42 paywall put up by DeGruyter at <a href="https://www.degruyter.com/document/doi/10.1515/jqas-2020-0012/html">https://www.degruyter.com/document/doi/10.1515/jqas-2020-0012/html</a>
</span></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;"> <span></span></span></p><a name='more'></a><p></p><i><span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">I don't
usually do academic reviews because it's usually unpaid labour for a corporation
with no benefit to me or my institution for doing it. I make rare exceptions
for papers that I would want to read and write about regardless, and for the
journal Meta-Psychology because I strongly support their mission. This paper was
relevant to my work at Sportlogiq, and I’m a fan of the authors</span></i>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><i><span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">Note that
neither the authors nor the reviewers were compensated by the publisher,
DeGruyter for their labour. Typically, authors will share preprint copies of their
manuscript if you contact them, but I can’t speak on behalf of individual authors.</span></i></p>
<p class="MsoNormal" style="line-height: normal; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;"><span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;"> <i>Also,
note that the published paper incorporates many of the revisions listed. These
comments are published here mostly to show the review process when there are
multiple rounds of revisions: A main review, and a follow up.</i></span></p>
<h3 class="MsoNormal" style="text-align: left;"><span style="font-family: "Times New Roman",serif;"> </span></h3>
<h3 class="MsoNormal" style="text-align: left;"><span style="font-family: "Times New Roman",serif;">Conflicts
of interest statement</span></h3>
<p class="MsoNormal"><span class="pagecontents"><span style="font-family: "Times New Roman",serif;">Possible
conflict 1: I work for a company (Sportlogiq) that has its own proprietary
method for assessing hockey shot quality, and I use that method in my own work.
However, I have no personal or professional incentive to either promote or
suppress this work.</span></span><span style="font-family: "Times New Roman",serif;"><br />
<br />
<span class="pagecontents">Possible conflict 2: Charmaine B. Dean was the
founding chair of the statistics department at Simon Fraser University, the
department in which did my PhD. However, she left for U of Western Ontario
around the same time I started there, so I still consider this to be an
arms-length relationship. <i>Because of my new appointment at U of Waterloo,
this is no longer an arms-length relationship, but it was at the time of the
review.</i></span></span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">My general
approach to reviews is to ask "what work does this need to be ready for
publication?", not to ask "is this worthy enough to be in
publication". As such, I rarely give a major mod / reject recommendation
to papers. If you have publication limitations, or a target rejection rate,
please take that into account. The paper is fine, it ticks all the boxes a
paper should. It's not important enough to rushed to the front of the queue
though.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;"> </span></p>
<h3 class="MsoNormal" style="text-align: left;"><span style="font-family: "Times New Roman",serif;">General
assessment</span></h3>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">"Algorithmically
Deconstructing Shot Locations as a Method for Shot Quality in Hockey" is
clear, well-written, and approachable despite using traditionally very
hard-to-explain methods like INLA. The tone, scope, and style make this paper a
good fit for JQAS, using the most recent editor's choice papers as a
comparison. The research itself is a meaningful addition to the assessment of
shot quality, specifically the characterization of shots into a small set of
profiles, and accounting for the handedness of the shoot.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">As I
understand the manuscript, non-negative matrix factorization (NMF) is the
method used to find these clusters, and a Log-Gaussian Cox Process (LGCP) is
used to spread shots and goals across a surface of locations that would
otherwise be too sparse and noisy to get useful information.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">Regarding
reproducibility, the data used is publicly available, and the method used is
described well enough that someone could recreate the analysis given the
dedication.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">Regarding
novelty, the work described isn't exactly groundbreaking, but it does
contribute to the general understanding of hockey dynamics, and the sports
analytics world is better off with this paper being published.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;"> </span></p>
<h3 class="MsoNormal" style="text-align: left;"><span style="font-family: "Times New Roman",serif;">Specifics
</span></h3>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">I have one
major and two minor issues with the manuscript, neither of which is a problem
with the research itself.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">My major
issue is that the results section is anemic. In the text, there are only two
paragraphs in a single subsection explicitly dedicated to results. The paper is
already short with 8 pages of writing outside of the captions and abstract.
Five of the six figures have high-quality results, along with good commentary
in the caption (the other figure is a diagnostic plot, it's fine too, but it's
not results). There's room for a lot more commentary in the main text.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">For
example: "Figure 3 shows the coefficient estimates for a few selected
players." This is a good opportunity to take more about these notable
players, either for their competence, their embodiment of an archetype like
"left-handed center", or for their uniqueness. Pick a few players and
tell the story that their shooting profiles reveal. Alternatively, what is it
the shooting percentage in each cluster?</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">My first
minor issue is an obscure problem with the data source, the NHL real time
tracking: It requires normalization across different eras. For example, in
January 2015, the NHL started using a new tracking system, and things like the
average physical distance between recorded events suddenly increased. See: <a href="https://www.nhl.com/news/nhl-sportvision-test-program-to-track-players-puck/c-750201">https://www.nhl.com/news/nhl-sportvision-test-program-to-track-players-puck/c-750201</a></span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">At the
beginning of the 2019-20 season, the distance to the net of shots suddenly
changed. Micah McCurdy of HockeyViz, a website already mentioned in the
manuscript, has commented on this change.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">My second
minor issue is with the use of the theoretical 'perfect player', which is the
distribution of all shots by players of a given position that resulted in
goals.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">The
authors mention that the top players have a different shot distribution than
the all-goals perfect player. The explanation given is that "There are
several strategies that players can adopt based on their strengths and
weaknesses; the perfect players only represent one strategy." Strategy
variation and game theory are one explanation, but there's something else:
survivor bias.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">The
perfect player represents all the shots with the best outcomes, but those kinds
of quality shots are not often available. A player, proficient scorer or not,
cannot often choose to take a shot from where they are or from somewhere closer
or at a better angle. Instead, when the opportunity to shoot presents itself, a
player can only choose or to trade that opportunity for the chance at a better
one. The perfect player represents the times when the best opportunities were
not just taken, but available.</span></p>
<p class="MsoNormal"><b><span style="font-family: "Times New Roman",serif;">I
recommend that this paper be published In JQAS with moderate priority and minor
revisions.</span></b></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;"> </span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;"> </span></p>
<h3 class="MsoNormal" style="text-align: left;"><span style="font-family: "Times New Roman",serif;">Follow Up comments
after a round of revisions</span></h3>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">In the
previous version of this draft, there were three issues that I wanted to see
addressed before giving a decision to 'accept': Major 1. A lack of specific
results (e.g. something a reader could use to describe it to a practitioner or
a coach), Minor 1. There are two possible data inconsistencies relating to the
NHL's event recording in early 2015 and early 2020. Minor 2. the overreach /
over simplification of the concept of a perfect player.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">The new
results Section 4.1 "Estimated Basis Functions" addresses Major Issue
1 by describing the shot characterization of two example players, how they
deviate from the theoretical perfect player, and the implications of this
deviation. Also, in Section 4.1 is a clarification about the strategic limitations
of the Perfect Player concept, thus addressing Minor Issue 2. Both issues are
addressed further in Section 4.2</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">Minor
Issue 1 was addressed by two comments in the concluding discussion.
Specifically, the effect of the data inconsistency in 2015 was checked, and
that the data used does not extend to the 2019-20 season with the other
inconsistency.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">Two new
issues have arisen: There is a broken ref on Page 9, Line 30 to "Figure
??". Also, the capitalization of "Perfect Player" is
inconsistent between Sections 4 and 5. I personally prefer it without
capitalization, but the most important thing is to be consistent. These can be
fixed in the copyediting stage.</span></p>
<p class="MsoNormal"><span style="font-family: "Times New Roman",serif;">Finally,
this revision has several other details that weren't in the original, and I
appreciate the thoroughness and rigour.</span></p>
<p class="MsoNormal"><b><span style="font-family: "Times New Roman",serif;">I am happy
to recommend that this manuscript be accepted as is.</span></b></p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-59027786420159977892021-10-19T12:07:00.003-07:002021-10-19T12:07:50.557-07:00Sampling, conditional probability, and random number generation<p class="MsoNormal" style="margin-bottom: 0cm;"> <!--[if gte mso 9]><xml>
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<![endif]-->Part of the motivation behind making
the course <a href="https://www.stats-et-al.com/2020/07/statistics-gambling-and-games-of-chance.html">Statistics and Gambling</a> is to infuse new applicability into introductory
or intermediate probability courses. This blog post is a look at how the course
is going to cover familiar probability topics with examples in games of chance,
and a simulation-based (rather than theory-based) approach.</p>
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number generation (RNG) in R, and applying RNG to demonstrate core concepts in
sampling, conditional probability, and conditional distributions. It is meant to
be a very surface-level primer on the topics, just enough to give context for
the deeper dives into specific games of chance.</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span></span></p><a name='more'></a><p></p><p class="MsoNormal" style="margin-bottom: 0cm;"><!--[if gte mso 9]><xml>
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<![endif]--></p><p class="MsoNormal" style="margin-bottom: 0cm;">
</p><p class="MsoNormal" style="margin-bottom: 0cm;">In this post, we cover<br /></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">1) Preliminary code</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">2) RNG (Random number generation)
in sampling</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">3) Sampling with and without
replacement</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">4) Conditional probability using
sampling</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">5) RNG from a distribution</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">6) Conditional probability using a
distribution</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">7) Conditional distributions</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">8) RNG and random seeds</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;">Preliminary Code</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Some of the code in this post
relies on the following data frame and function.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">card_rank
=
rep(c(2:10,"Jack","Queen","King","Ace"),
times=4)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">card_suit
=
rep(c("Clubs","Hearts","Diamonds","Spades"),
each=13)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">card_name
= paste(card_rank, card_suit, sep=" of ")</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">df_cards_52
= data.frame(card_rank, card_suit, card_name)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">names(df_cards_52)
= c("rank","suit","name")</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">n.of.each
= function(x)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>out = rev(sort(as.numeric(table(x))))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>return(out)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;">RNG using sampling</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><i>The set.seed() command ensures
that the results you get are the same as those written here. Details about
set.seed() are given in the optional section on random seeds at the end of this
post.</i></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">This code samples all 5 numbers
from 1 to 5 in a random order using R’s internal random number generator (RNG).
</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(8675309)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 4 3 2 1 5</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">However, you can't sample 6
numbers from a population of 5 values without replacement. Replacement is explained
after this section.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=6)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: Error, cannot take a sample larger than the population when 'replace =
FALSE'.</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We can do it if we sample with
replacement, where each number is put back into the pool after we sampled it.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=6, replace=TRUE)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 5 3 1 5 3 1</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">By default, every value in x is
equally likely to be selected, but we can change that with the prob setting.
The following code samples from the number 1 80% of the time, and 2, 3, 4, or 5
5% of the time each.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=20, prob=c(.800, .05, .05, .05, .05), replace=TRUE)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 1 1 3 1 1 3 3 1 1 1 1 1 1 3 2 3 1 1 1 1</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The probabilities don't need to
add up to 1 if you prefer to use weights instead, like this, code, were 1 is
selected 4 times as often as 2, 3, 4, or 5.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=20, prob=c(4,1,1,1,1), replace=TRUE)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 1 1 1 4 2 2 3 5 1 1 1 1 1 5 5 1 5 1 4 1</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We can also use it to sample from
things that aren't numbers in a range. The following code sets up a deck of 52
cards and draws 5 cards from the deck.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">hand
= sample(x= df_cards_52$name, size=5)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">hand</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: "2 of Diamonds"<span style="mso-spacerun: yes;">
</span>"7 of Clubs"<span style="mso-spacerun: yes;"> </span>"7
of Spades"<span style="mso-spacerun: yes;"> </span>"Queen of
Clubs" "10 of Clubs"</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;">Sampling with and without replacement</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">There are two fundamental ways to
sample. Both ways start out the same: select an object from a population
randomly (e.g., select a person from the residents of a city, or a card from a
deck).</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">If you are sampling with
replacement, you include place that object back into the sample, so that they
same object can be sampled again later (e.g., you put the card back into the
deck after selecting it).
</p><p class="MsoNormal" style="margin-bottom: 0cm;">If you ware sampling without
replacement, you remove the object from the sample so that every object can be
sampled either once or not at all (e.g., You keep the card in your hand after
selecting it. After you call someone for a poll you make a note not to call
that phone number again).
</p><p class="MsoNormal" style="margin-bottom: 0cm;">This block of code samples from
the numbers 1 to 5 without replacement. If you don't specify, the sample()
function will always sample without replacement.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(1:5,
size=5, replace=FALSE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(1:5,
size=5, replace=FALSE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(1:5,
size=5, replace=FALSE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; tab-stops: 92.25pt;"><span style="font-family: "Courier New";"># Output:<span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] 3 4 2 5 1</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] 2 1 3 5 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] 2 5 3 4 1</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Notice how after each line, the
numbers 1 to 5 each appear exactly once. If we were to sample with replacement,
as we do in the following code block, sometimes numbers appear more than once
in a line.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(1:5,
size=5, replace=TRUE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(1:5,
size=5, replace=TRUE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(1:5,
size=5, replace=TRUE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Output:</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] 3 2 4 2 5</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] 3 2 3 2 1</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] 1 4 4 2 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Most games with cards or tiles use
sampling without replacement because multiple cards are drawn before shuffling
the deck or tiles. Most games with dice use sampling with replacement because
the result of any one die roll does not affect the result of other dice rolls.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>Examples of games that use
sampling without replacement include:
<p class="MsoNormal" style="margin-bottom: 0cm;">1) <a href="https://www.stats-et-al.com/2019/04/bingo-analysis-tutorial-in-r.html">Bingo</a>, which samples from a set
of 75 balls.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">2) Texas Hold'Em Poker, which
samples from a deck of 52 cards.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">3) Pai Gow Poker, which samples from
a deck of 53 cards (52 + 1 joker).</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">4) Mahjong, which samples from a
set of 144 tiles or cards.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Examples of games that use
sampling with replacement include:</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">5) Craps, which involves rolls of
five six-sided dice repeatedly.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">6) Yahtzee, which involves rolls of
two six-sided dice repeatedly.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">7) <a href="https://www.stats-et-al.com/2020/12/borel-dice-edition-brute-forcing.html">Borel</a>, which involves rolls of
multiple variable-sided dice repeatedly.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">8) <a href="http://mathcentral.uregina.ca/RR/database/RR.09.00/treptau1/game4.html">Hubbub</a>, which involves flipping
five two-sided dice repeatedly.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">There are some activities can be
treated as somewhere between sampling with and without replacement, these
include:</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">9) Blackjack, which typically draws
cards from a 'shoe' of several decks shuffled together, effectively making one
large superdeck of hundreds of cards. A superdeck of ten decks, or 520 cards,
is large enough that strategies that rely on sampling without replacement, like
counting cards, are rendered much less effective.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">10) <a href="https://www.stats-et-al.com/2021/01/lottery-tickets-baseball-cards-and.html">Buying randomized packs of collectible cards</a>, collectible stickers, or trading cards. By buying cards in a
pack or box, you are sampling without replacement from a population of
thousands or millions of printed cards or stickers - a population large enough
that the difference between sampling with or without replacement doesn't
matter. Also, packs of collectibles have a more complicated probability
distribution.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;">Conditional probability using sampling</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>Imagine this situation: You draw
one card from a standard deck of 52 cards - it's an ace. What is the chance
that the second card from your deck will also be an ace?
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">One way to solve this is by
counting.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">- There are 4 aces in a deck of 52
cards.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">- After drawing the first ace,
there are 3 aces left in a deck of 51 cards.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Pr(2nd ace | 1st ace) = 3/51 = 1/17
= 0.05824</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The vertical bar means 'if', or
'given'.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">Counting methods work well for
simple cases where counting principles can be applied easily. For moderately
complicated card situations, we can also use combinatorics, which are covered
elsewhere. We can also use Monte Carlo, or simulation-based methods to estimate
things. Counting and combinatorics can provide exact answers to simple and
moderately complex probability problems, Monte Carlo methods can provide
approximate answers to very complex probability problems (for example, if some
cards were more likely to appear than others, for some reason).
</p><p class="MsoNormal" style="margin-bottom: 0cm;">The following code block estimates
the conditional probability of drawing a second ace from a deck after drawing
the first ace. It's overkill, but it shows the concept on a problem that we
already know the answer to.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">First, we simulate drawing a
two-card hand many times.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nsims
= 50000</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">all_hands
= matrix("", nrow=Nsims, ncol=2)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">for(k
in 1:Nsims)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>all_hands[k,] = sample(df_cards_52$rank, size=2)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Next, define event A as
"first card is an ace", and event B as "second card is an
ace", and estimate the probability of event B given event A.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">count_A
=<span style="mso-spacerun: yes;"> </span>length(which(all_hands[,1] ==
"Ace"))</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">count_AandB
= length(which(all_hands[,1] == "Ace" & all_hands[,2] ==
"Ace"))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_BifA<span style="mso-spacerun: yes;"> </span>= count_AandB / count_A</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nsims<span style="mso-tab-count: 2;"> </span><span style="mso-tab-count: 1;"> </span>#Output:
50000</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">count_A<span style="mso-tab-count: 2;"> </span>#Output: 3719</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">count_AandB<span style="mso-tab-count: 1;"> </span>#Output: 216</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_BifA<span style="mso-tab-count: 2;"> </span>#Output: 0.05808</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">To reiterate, we drew 50000
two-card hands, and of those 3719 had an ace as the first card, and of those
3719 first-ace hands, 216 of them had two aces. From those numbers we can
estimate the conditional probability as 216/3719 = 0.05808, which is close to,
but not exactly the true value of 0.05824.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We could use these event counts to
estimate the probability of the first card being an ace as 3719 / 50000, or
0.07438, as compared to the true value of 4/52 = 1/13 = 0.7692.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">We could also estimate the chance of
both cards of being aces as 216 / 50000 = 0.00432, as compared to the true
value of 1/221 = 0.00452.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">By increasing the number of
simulations Nsims, we could increase the accuracy, but it would take longer to
calculate.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">Conditional probability is calculated
by pr(B|A) = pr(A and B) / pr(A). We get the conditional probability by
estimating the base probabilities using counts as well, which is what the
following code block does.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_A
<span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>=
length(which(all_hands[,1] == "Ace")) / Nsims</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_AandB
= length(which(all_hands[,1] == "Ace" & all_hands[,2] ==
"Ace")) / Nsims</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_BifA<span style="mso-spacerun: yes;">
</span>= pr_AandB / pr_A</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_A<span style="mso-tab-count: 2;"> </span>#
0.07438</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_AandB<span style="mso-tab-count: 1;"> </span>#
0.00432</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_BifA<span style="mso-tab-count: 1;"> </span># 0.05808</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p><p> </p><p><span style="font-size: 20.0pt; line-height: 107%;"></span><span style="font-size: 20.0pt; line-height: 107%;">RNG from a distribution</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>RNG can do a lot more than sample
from a pre-made set of values. We can also use it to pick a value (or many
values) from a probability distribution. Commonly used distributions include
the uniform, Poisson, geometric, and binomial, but there are many, many more
distributions that can be used.
<p class="MsoNormal" style="margin-bottom: 0cm;"><br /></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">par(mfrow
= c(2,2))</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">barplot(dunif(1),
xlim=c(0, 1.5), main="Uniform, min=0, max=1",
ylab="Density")</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">barplot(dbinom(x=0:10,
size=10, prob=0.5), main="Binomial, n=10, p=0.5",
ylab="Density")</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">barplot(dpois(x=0:10,
lambda=5), main="Poisson, lambda=5", ylab="Density")</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">barplot(dgeom(x=0:10,
prob=0.25), main="Geometric, n=10, prob=0.25",
ylab="Density")</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">par(mfrow
= c(1,1))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJ2PdSmd5KSJy-ovFqaeoW2UxYvF2FNUFZoyQxXO5K0twCv_rK7Y50VQ_NgJ6J34XqKtXrsC3rBb78NqB2TGKs3R-STWkJ7eODIXvC6E6ozt3nvlEYthKHpodP0Jg725qj0CQCbkW3rQ5d/s1105/4+Distributions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="4 Distributions: Uniform, Binomial, Poisson, and Geometric. Stats-et-al.com" border="0" data-original-height="664" data-original-width="1105" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJ2PdSmd5KSJy-ovFqaeoW2UxYvF2FNUFZoyQxXO5K0twCv_rK7Y50VQ_NgJ6J34XqKtXrsC3rBb78NqB2TGKs3R-STWkJ7eODIXvC6E6ozt3nvlEYthKHpodP0Jg725qj0CQCbkW3rQ5d/w400-h240/4+Distributions.png" width="400" /></a></div><p></p><p class="MsoNormal" style="margin-bottom: 0cm;"><!--[if gte mso 9]><xml>
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<style>
/* Style Definitions */
table.MsoNormalTable
{mso-style-name:"Table Normal";
mso-tstyle-rowband-size:0;
mso-tstyle-colband-size:0;
mso-style-noshow:yes;
mso-style-priority:99;
mso-style-parent:"";
mso-padding-alt:0cm 5.4pt 0cm 5.4pt;
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mso-para-margin-right:0cm;
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mso-para-margin-left:0cm;
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font-family:"Calibri",sans-serif;
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mso-ascii-theme-font:minor-latin;
mso-hansi-font-family:Calibri;
mso-hansi-theme-font:minor-latin;
mso-bidi-font-family:"Times New Roman";
mso-bidi-theme-font:minor-bidi;
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<![endif]--></p><p class="MsoNormal" style="margin-bottom: 0cm;">
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 18.0pt; line-height: 107%;">Uniform</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The function 'runif' is short for
'Random from UNIForm distribution'. This is the most basic of random number
generation, and we won't be using it directly very much.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">This generates a random number
between 0 and 1.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">runif(n=1)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 0.7209039</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">This generates 10 random numbers
between 0 and 1.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">runif(n=10)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output (rounded): [1] 0.876 0.761 0.886 0.456 0.166 0.325 0.509 0.728 0.990
0.035</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">This generates a random number
between 3 and 5.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">runif(n=1,
min=3, max=5)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 3.304747</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">You can use runif() to produce
random events of a known probability. For example, this is a simple coin flip
program.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">coin
= "heads"</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">if(runif(1)
< .5){coin = "tails"}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">coin</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] "heads"</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 18.0pt; line-height: 107%;">Binomial</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The function 'rbinom' is short for
'Random from BINOMial distribution'. This is used to count the number of
'successes' in a fixed number of trials. (e.g., the number of goals from a
fixed number of shots, or the number of wins in a season of fixed length).</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We can make an even shorter coin
flip program with rbinom</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rbinom(n=1,
size=1, prob=0.5)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 0</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Or flip 10 coins and get the
results from each one.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rbinom(n=10,
size=1, prob=0.5)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 0 0 0 0 0 1 0 0 1 1</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Or flip 10 coins all at once and
count the number of heads.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rbinom(n=1,
size=10, prob=0.5)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output:</span> <span style="font-family: "Courier New";">[1] 6</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">In rbinom, n determines the number
of values to generate, size determines the number of trails, and prob
determines the probability of success. A success can be defined as anything
binary, or a yes/no thing.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">This RNG code treats rolling a 6
on a 6-sided die as a success (prob = 1/6), two dice are rolled (size = 2), and
this is repeated 10 times (n = 10).
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rbinom(n=10,
size=2, prob=1/6)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 0 1 0 0 0 1 0 0 0 0</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Similarly, this code generates the
number of wins in 10 seasons of 162 games of an above-average Major League
Baseball team, with a 55% win chance.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rbinom(n=10,
size=162, prob=0.55)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1]<span style="mso-spacerun: yes;"> </span>91 102<span style="mso-spacerun: yes;"> </span>95<span style="mso-spacerun: yes;">
</span>90<span style="mso-spacerun: yes;"> </span>84<span style="mso-spacerun: yes;"> </span>82<span style="mso-spacerun: yes;">
</span>89<span style="mso-spacerun: yes;"> </span>81<span style="mso-spacerun: yes;"> </span>93<span style="mso-spacerun: yes;"> </span>86</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">This does the same thing for a
below-average team.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rbinom(n=10,
size=162, prob=0.45)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 70 69 67 67 69 86 80 80 66 62</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 18.0pt; line-height: 107%;">Poisson</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>The Poisson distribution is used
to count events in a fixed time. (e.g, goals in a soccer game). A typical
English Premier League soccer game has 1.2 goals per team, so we can generate
the goals an average team gets in 10 games with the following code.
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rpois(n=10,
lambda=1.2)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 0 0 1 2 1 2 0 3 2 0</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The average NHL team scores about
2.85 goals in regulation.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rpois(n=10,
lambda=2.85)</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 3 4 3 4 4 1 2 2 3 7</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 18.0pt; line-height: 107%;">Geometric</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The geometric distribution counts
the number of successes until the first failure (or the number of failures
until the first success).</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The lengths of 10 winning streaks
of a team with a 70% winning chance (and a 30% chance of breaking the streak)
in any given game can be generated with this code.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">If you want to count the number of
wins in a 'streak', use this:
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rgeom(n=10,
prob=0.3)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 0 0 5 4 0 2 0 1 3 3</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">If you want include the number of
games, which includes the loss that ends the streak, add a +1 to the end of the
code like this:
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rgeom(n=10,
prob=0.3) + 1</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 1 3 3 1 2 1 1 3 5 2</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The chance of getting two aces
from a two-card hand (i.e., pocket aces) is 1/221. The number of hands you
would have to wait before getting the pocket aces, not including the hand with
the pocket aces, can be simulated with
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rgeom(n=10,
prob=1/221)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span><span style="font-family: "Courier New";">#
Output: [1] 229 254<span style="mso-spacerun: yes;"> </span>57 410<span style="mso-spacerun: yes;"> </span>27 103 133 184 262 374</span>
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;">Conditional probability using distributions</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>We can use RNG from distributions
to find conditional probability too.
<p class="MsoNormal" style="margin-bottom: 0cm;">Consider a hypothetic game between
two soccer teams the Tigers and Lions. We'll define two events.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Event A: The Tigers score 2 or
more goals.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">Event B: The Tigers win (i.e, they
score more than the Lions).</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We can find the probability these
events (Pr(A) and Pr(B)), and use that to find those to find some conditional
probabilities, (Pr(B given A) and Pr(A given B))
</p><p class="MsoNormal" style="margin-bottom: 0cm;">We start by generating goals
scored for a large number of games between each team. Assume for now each team
is average, and that they score an average of 1.2 goals per game. Other
possible scenarios are considered in the exercises.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nsims
= 100000 # The number of games simulated</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">g_tigers
= rpois(Nsims, lambda=1.2)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">g_lions<span style="mso-spacerun: yes;"> </span>= rpois(Nsims, lambda=1.2)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Next, we estimate probabilities
based on the proportion of simulations that fit the events.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_win<span style="mso-spacerun: yes;"> </span>= length(which(g_tigers > g_lions)) /
Nsims</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_draw
= length(which(g_tigers == g_lions)) / Nsims</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_loss
= length(which(g_tigers < g_lions)) / Nsims</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_score2
= length(which(g_tigers >= 2)) / Nsims</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_A = pr_score2</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_B = pr_win</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="mso-ansi-language: PT-BR;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Next, we estimate the probability
of the intersection, Pr(A and B), which is the case where both events happen.
In this case, that's where the Tigers score 2 or more goals AND win. Since
these two events aren't independent (scoring more makes a team more likely to
win), we can't just multiply the two probabilities. </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_AB
= length(which(g_tigers >= 2 & g_tigers > g_lions)) / Nsims</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">From that we can get conditional
probabilities, like Pr(Tigers win IF they score 2 or more), and Pr(Tiger score
2 or more IF they win)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_BifA = pr_AB / pr_A</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_AifB = pr_AB / pr_B</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_A <span style="mso-tab-count: 2;"> </span>#
Output: 0.33655</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_B<span style="mso-tab-count: 2;"> </span>#
Output: 0.36205</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_AB<span style="mso-tab-count: 2;"> </span>#
Output: 0.25257</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">pr_AifB<span style="mso-tab-count: 1;"> </span>#
Output: 0.6976108</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">pr_BifA<span style="mso-tab-count: 1;"> </span># Output: 0.7504680</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">To recap, the Tigers are estimated
to have a 33.7% chance of winning, and the Lions have an estimated 36.2% chance
of winning. In theory, these two numbers should be the same because the distributions
that generated them are the same, so the simulation approach is showing its weaknesses.
The two probabilities don’t add to 100% because there is a large chance of a
draw/tie where neither team wins.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The chance of the Tigers winning
AND scoring two or more is 25.2%.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">Also, when the Tigers score 2 or
more, they win 69.8% of the time.
</p><p class="MsoNormal" style="margin-bottom: 0cm;">Likewise, when the Tigers win,
75.0% of the time they do so by scoring 2 or more.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;">Conditional distributions</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The number of goals in sports like
hockey, soccer, and lacrosse fundamentally depend on two things: The number of
shots taken, and the goalie's general ability to stop shots. There are lots of
other factors involved, but for the sake of simplicity, let's assume for now
that the number of shots a goalie faces follow a Poisson distribution, and that
goals follow a binomial distribution. That is, every shot is independent and
has the same change of going in the net.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">For NHL hockey, ignoring overtime,
the average team takes about 30 shots per game, and about 9.5% of those shots
become goals. (For an average of 2.85 goals per game)
</p><p class="MsoNormal" style="margin-bottom: 0cm;">We could simulate the number of
goals of an average team using rpois() and rbinom()
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span></p><span style="font-family: "Courier New";">Nsims
= 100000</span>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nshots
= rpois(n=Nsims, lambda=30)</span></p><span style="font-family: "Courier New";">Ngoals
= rbinom(n=Nsims, size=Nshots, prob=.095)</span>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">summary(Ngoals)
# Mean, Median, Quartiles</span></p><span style="font-family: "Courier New";">#
table(Ngoals) / Nsims # Probability distribution of goals</span>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Output:</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Min. 1st Qu.<span style="mso-spacerun: yes;">
</span>Median<span style="mso-spacerun: yes;"> </span>Mean 3rd Qu.<span style="mso-spacerun: yes;"> </span>Max. </span></p><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>0.000<span style="mso-spacerun: yes;">
</span>2.000<span style="mso-spacerun: yes;"> </span>3.000<span style="mso-spacerun: yes;"> </span>2.855<span style="mso-spacerun: yes;">
</span>4.000<span style="mso-spacerun: yes;"> </span>13.000 </span>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">mean(Ngoals)
# 2.855, close to 20 * 0.095</span></p><span style="font-family: "Courier New";">var(Ngoals)
# 2.872, close to the mean</span>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We could filter these simulations
to estimate conditional distributions. For example: What is the average number
of goals scored when the team gets 25 shots?</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#summary(Ngoals[Nshots
== 25])</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">mean(Ngoals[Nshots
== 25]) # Output: 2.353</span></p><span style="font-family: "Courier New";">var(Ngoals[Nshots
== 25]) # Output: 2.110, less than the mean.</span>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We would expect the answer to be
close to 25 * 0.095.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">25
* 0.095 # Output: 2.375</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">What is the mean and variance of
the number of goals conditional on there being 40 or more shots?</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#summary(Ngoals[Nshots
>= 40])</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">mean(Ngoals[Nshots
>= 40]) # Output: 3.998</span></p><span style="font-family: "Courier New";">var(Ngoals[Nshots
>= 40]) # Output: 3.499, less than the mean again.</span>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We can filter in the opposite direction
too. What is the distribution of the number of shots taken in 5-goal games?</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">summary(Nshots[Ngoals
== 5])</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Output:</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
<span style="mso-spacerun: yes;"> </span>Min. 1st Qu.<span style="mso-spacerun: yes;"> </span>Median<span style="mso-spacerun: yes;">
</span>Mean 3rd Qu.<span style="mso-spacerun: yes;"> </span>Max. </span></p><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>15.0<span style="mso-spacerun: yes;">
</span>29.0<span style="mso-spacerun: yes;"> </span>32.0<span style="mso-spacerun: yes;"> </span>32.2<span style="mso-spacerun: yes;">
</span>36.0<span style="mso-spacerun: yes;"> </span>52.0</span>
<p class="MsoNormal" style="margin-bottom: 0cm;">Finally, we can visualize all this
with a side-by-side boxplot. Each column represents the distribution of shots
taken conditional on the number of goals. The coloured box in each column is
the middle half of that distribution, so in our simulation, the 50% of shutouts
(zero-goal games) had between 25 and 31 shots. The number of shots in a game
increases with the number of goals, but a lot less than you might expect.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">boxplot(Nshots
~ Ngoals)</span></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxF-jB8GjxKlexs7VkW0KriodiNEbVrjmb8gWucP7zfeFu6QaeCzSNzeZN1eN-BeMdTwP50Sqct2Eik7CZ08bzggdW9ngAwuhx7FbYQdtUU8nqitzmuoToHHlznKc1Pq2to2x1R5AI6WNe/s665/Nshots+Ngoals.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Number of shots (x) by number of goals (y), stats-et-al.com" border="0" data-original-height="664" data-original-width="665" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxF-jB8GjxKlexs7VkW0KriodiNEbVrjmb8gWucP7zfeFu6QaeCzSNzeZN1eN-BeMdTwP50Sqct2Eik7CZ08bzggdW9ngAwuhx7FbYQdtUU8nqitzmuoToHHlznKc1Pq2to2x1R5AI6WNe/w400-h400/Nshots+Ngoals.png" width="400" /></a></div><p></p><p class="MsoNormal" style="margin-bottom: 0cm;"><br /></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="font-family: "Courier New";"><!--[if gte mso 9]><xml>
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</span></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;">(Optional) RNG and random seeds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The random number generator (RNG)
in R is not truly random. Like any computer without specialized hardware or
some external randomness source, R uses a pseudo random number generator. The
randomness is governed by a seed.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">By default, R uses the exact time
in milliseconds of the computer's clock when R starts up to generate a seed.
(Slot machines and card shuffling in online poker software usually use similar
methods for their pseudo random number generators.)
</p><p class="MsoNormal" style="margin-bottom: 0cm;">We can change this seed in two
ways. First, we can force R to use a particular random seed of our choosing. In
the following code block, we set an arbitrary seed before running code that
uses randomness. Notice that when we use the same seed, we get the same
results. The results will be the same even if the code is run on a different
computer.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(1337)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 2 3 5 1 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 2 4 3 5 1</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) #</span> <span style="mso-spacerun: yes;"> </span><span style="font-family: "Courier New";">Output: [1] 5 2 1 3 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(1337)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 2 3 5 1 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 2 4 3 5 1</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) #</span> <span style="mso-spacerun: yes;"> </span><span style="font-family: "Courier New";">Output: [1] 5 2 1 3 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(420)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 5 1 2 3 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 1 5 4 3 2</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 1 5 4 3 2</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> <br /></p><p class="MsoNormal" style="margin-bottom: 0cm;">In the following block of code, we
force R to pick a seed based on the computer's clock. Notice that running the
same code gets different results each time.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(NULL)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 2 4 5 1 3</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 2 3 5 1 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 2 1 5 3 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(NULL)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 3 4 5 2 1</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 2 5 1 3 4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sample(x=1:5,
size=5) # Output: [1] 5 1 3 4 2</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p><p class="MsoNormal" style="margin-bottom: 0cm;">In the code blocks in the rest of
this section, set.seed(12345) is placed at the beginning of each block so that
anyone can recreate the same results even though randomness is used. Remove it
or use set.seed(NULL) to produce your own unique results.
</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 20.0pt; line-height: 107%;">Exercises</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Estimate the probabilities pr(A), pr(B),
and pr(A and B), and pr(A|B), and pr(B|A) from the “Tigers vs. Lions” example in
“conditional probability using distributions section” for the following
situations.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">A) Tigers are a better than the
Lions. (Tigers score an average 1.4 goals/game, and Lions score an average 1.0
goals/game).</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">B) Tigers are worse than the
Lions. (Tigers average 1.0 goals/game, Lions average 1.4 goals/game)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">C) Tigers and Lions are both very
attack-heavy teams (Tigers and Lions both average 2.0 goals/game)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">D) Tigers and Lions are both very defense-heavy teams (Tigers and Lions both average 0.5 goals/game)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">See also:</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><a href="https://www.educba.com/random-number-generator-in-r/">https://www.educba.com/random-number-generator-in-r/</a></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><a href="http://www.cookbook-r.com/Numbers/Generating_random_numbers/">http://www.cookbook-r.com/Numbers/Generating_random_numbers/</a></p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p><p class="MsoNormal" style="margin-bottom: 0cm;"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXR7NJ4U_a56m_ZVH4HoqQ3Xo7TgAGSC-KT3NCfo-6VptqLIVqIC1FOU3UhEXPhuRcP0_KoVVdjtBs448ypPAU7BpoW6g98DlNWQuaZrz5vqahCzpWhQYYU-by7RGzrePWdFb1atOFkfzw/s2048/Chica+Sprawl+2021.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="A chihuahua yorkie sprawls belly up across way too much of the couch." border="0" data-original-height="2048" data-original-width="1536" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXR7NJ4U_a56m_ZVH4HoqQ3Xo7TgAGSC-KT3NCfo-6VptqLIVqIC1FOU3UhEXPhuRcP0_KoVVdjtBs448ypPAU7BpoW6g98DlNWQuaZrz5vqahCzpWhQYYU-by7RGzrePWdFb1atOFkfzw/w300-h400/Chica+Sprawl+2021.jpg" width="300" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Bellyrub distributions are not so random.</td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td></tr></tbody></table><br /> <br /></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-27428135372229015302021-06-26T17:52:00.016-07:002021-07-20T13:49:29.165-07:00The Bottleneck Retirement Plan<p class="MsoNormal"><!--[if gte mso 9]><xml>
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</xml><![endif]--> I do not have a voluntary retirement plan or a pension. I
have the means to put money away specifically for retirement, but I choose not to. Instead, I use an investment strategy that has been described as "the most and least insane thing I've ever heard". Here is that strategy:<br /></p>
<p class="MsoNormal"> <span></span></p><a name='more'></a><p></p>
<p class="MsoNormal">In 2017, I called this deliberate non-plan the "Smith
& Wesson Retirement Plan". In the case of a climate Armageddon, a
pension is worthless. In the case a post-scarcity utopia, a pension is
unnecessary.<span style="mso-spacerun: yes;"> Factoring in some totally made up probabilities, </span>I estimated back then that
$1 locked away for 35 years was worth about $0.60, assuming it was a retirement
plan with traditional investments. My valuation has not improved since then.</p>
<p class="MsoNormal">Tax deferment and generous employee contribution matching
aren't enough to make up the difference, especially if you can negotiate for a
raise in lieu of pension matching.
</p><p class="MsoNormal">To be clear, this valuation is specifically about retirement schemes
in which money is locked away until some arbitrary retirement age, usually 65,
in exchange for a one-time payout or a constant income until death. </p><p class="MsoNormal">Today, I'd like to update that philosophy into something less grim:
<b>The Bottleneck Retirement Plan</b>.
</p><p class="MsoNormal"> </p><p class="MsoNormal">The goal of the Bottleneck Plan is traditional: maximize the
chance of a comfortable retirement. However, unlike a traditional plan, it
acknowledges the probability that you and everyone you love could boil to death in a
hell of our own making.
</p><p class="MsoNormal"> The Bottleneck Plan investment strategy to achieve its goal
is simple: Invest heavily in a wide portfolio of companies that are all-in on
clean energy and technology.
</p><p class="MsoNormal"> </p>
<p class="MsoNormal">The Plan works on the assumption that at least one of two
things will happen:</p>
<p class="MsoNormal"> 1. Clean energy and environmental services see massive
growth and long-term gains.</p>
<p class="MsoNormal">2. Society collapses, taking your retirement portfolio with
it.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">It's possible that both will happen, in sequence. It's practically
impossible that neither will happen.</p>
<p class="MsoNormal"> </p>
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh069S8S8qEuzkv6Oz9PmX7Vn2-vyEdRWtfmPXAkISdOxruoBxyiwWhBbrmQH9p5vGk9T97_xasl85_3vvQSc0mrJdEhVAuFc120Cybxjys2rK-Qe8b4njVEOSAY9PgSvHi7BaKj3oww5IR/s913/Bottleneck+Plan.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="428" data-original-width="913" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh069S8S8qEuzkv6Oz9PmX7Vn2-vyEdRWtfmPXAkISdOxruoBxyiwWhBbrmQH9p5vGk9T97_xasl85_3vvQSc0mrJdEhVAuFc120Cybxjys2rK-Qe8b4njVEOSAY9PgSvHi7BaKj3oww5IR/s320/Bottleneck+Plan.png" width="320" /></a></div><p class="MsoNormal">This assumption does not apply to every company, or even
every sector, but to the field as a whole. It doesn't mean that anybody with a
leaf in their logo is guaranteed money, or even that any technological path is
guaranteed money. It simply means that the field of clean tech (clean energy,
carbon capture, and environmental restoration) can be collectively is as close
to guaranteed to provide you old age income as any investment can possibly be,
including cash. It also does not mean that other investments will fail in the
long-term.</p>
<p class="MsoNormal">In 2020, I wrote about using such investments as a means to 'buy
absolution' for your climate sins. It may quell your climate anxiety, but whether or not you making
these investments actually helps in some way is a more complicated question.
Dollar-for-dollar carbon offsets are probably still your best shot at that, but
you are just giving away money. Buying $10,000 in clean energy stock is a lot easier
to justify to your family than buying $1,000 in offsets.</p>
<p class="MsoNormal">Wealthsimple has a 'clean energy' category, which is a good
start, but they try to discourage risky and day-trader style behaviour, so it's
mostly mid and large caps (larger, better established companies that have a
larger total value, or market capitalization). You can find a few more small
caps with the text search. Also, every company wants to appear green, so do
some due diligence and find if a company's main source of revenue is clean
tech, or if it's just a small part of their operations for marketing purposes.<br /><br />You may notice that I'm only minimally invested in solar power manufacturers, that's just because those investments are obvious choices and I expect they have already reached fair value. Impossible meats isn't on this list either because it's been hyped up already. Most of these are smaller, high-risk companies that only produce a narrow range of products.<br /></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b><span lang="PT-BR" style="mso-ansi-language: PT-BR;">AQMS*,
Aqua Metals</span></b><span lang="PT-BR" style="mso-ansi-language: PT-BR;">,
NASDAQ, USD.<span style="mso-spacerun: yes;"> </span></span>This company
recycles lead, mostly by extracting it from wet cell car batteries. Solar
panels use a lot of lead, so there's opportunity here for recycling old panels
and for making new ones sustainably.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>AQN*, Algonquin Power & Utilities</b>, TSE, CAD. A
collection of diverse clean energy and wastewater collection projects. Some
hydroelectric. Also, some natural gas, unfortunately. Not actually owned by the
Algonquin people as far as I can tell. (Please correct me on this if I'm wrong)</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>BEP, Brookfield Renewable Partners</b>, NYSE, USD.
Diverse generation of clean energy, including cogeneration. No natural gas.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>BLDP, Ballard Power</b>, TSE/NYSE, CAD/USD. Manufacturers
of hydrogen fuel cell and backup power systems.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>CLMT, Purpose Global Climate Opportunities Fund</b>, ETF,
CAD. Exchange traded fund focusing on clean energy.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b><span lang="PT-BR" style="mso-ansi-language: PT-BR;">CSIQ,
Canadian Solar</span></b><span lang="PT-BR" style="mso-ansi-language: PT-BR;">,
NASDAQ, USD. </span>Manufacturers of solar cells.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>GRN*, Greenlane Renewables</b>, TSE-V, CAD. Collecting
biogas from landfills and poop, and converting it into renewable natural gas
for electricity and fuel.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>INE*, Innergex Renewable Energy</b>, TSE, CAD. A
collection of diverse clean energy projects. No natural gas. Some
hydroelectric.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>LPS*, Legend Power Systems</b>, TSE-V, CAD. Manufacturers
of a consumer-side smart grid system that reduces total power usage and
protects against surges.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>NPI, Northland Power</b>, TSE, CAD. Diverse generation of
clean energy, including cogeneration and "clean burning" natural gas.
</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>OPTT*, Ocean Power Technologies</b>, USD. Manufacturers
of wave power generators. Honestly, this is a real long shot, and I don't know
if they will ever be profitable.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>PYR*</b>, PyroGenesis, TSE, CAD. Manufacturers of plasma
torches for extracting metal from waste.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b><span lang="PT-BR" style="mso-ansi-language: PT-BR;">RNW*</span></b><span lang="PT-BR" style="mso-ansi-language: PT-BR;">, TransAlta Renewables, TSE, CAD. </span>The
renewables branch of TransAlta. Power purchase and distribution of clean energy
and natural gas.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>SOL, ReneSola</b>, NYQ, USD. Solar power manufacture and
power generation.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>SOLR*, Solar Alliance Energy</b>, TSE-V, CAD. Development
and procurement of solar power plants. Very small cap, not yet profitable.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>SUNW, Sunworks</b>, NASDAQ, USD, Manufacturers of solar
cells.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>XBC*, Xebec Adsorption</b>, TSE, CAD. Generation of
renewable natural gas from biogas, and electrolysis from hydrogen.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>PLUG, Plug Power</b>, NYSE, USD. Hydrogen fuel cell
manufacturers. </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>VEGN, US Vegan Climate ETF</b>, ETF, USD. Exchange traded
fund focusing on climate solutions.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><i>At the time of writing, the author holds a long position
in any stock with an asterisk * next to its symbol (which means he personally
profits when these stocks go up). </i></p>
<p class="MsoNormal"><i>The author has completed the Canadian Securities Course,
but the credential has expired. This is not financial advice. Copyright not
intended. Those experiencing diamond hands for more than 4 hours should consult
a doctor. All reproductions or accounts without the express written consent of
Major League Baseball...</i></p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-21100980150974038282021-04-11T18:25:00.014-07:002021-04-11T18:35:00.242-07:00Wow, what are the odds? (Part 1: American Odds, Decimal Odds, and Implied Probability)<!--[if gte mso 9]><xml>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 5"/>
<w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Closing"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Signature"/>
<w:LsdException Locked="false" Priority="1" SemiHidden="true"
UnhideWhenUsed="true" Name="Default Paragraph Font"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Message Header"/>
<w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Salutation"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Date"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Block Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/>
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<p class="MsoNormal">The term "odds" is slippery because it's used to
mean different things in different contexts. In layperson terms,
"odds" is often used as a synonym for probability. In proper statistical
terms, "odds" is a function of probability, but it's not the same as
probability. There are also other uses of the term "odds" in gambling
contexts which are functions of a parallel concept called "implied probability".
In these notes, we're going to look at some common types of odds in statistics and
gambling contexts, and some of the calculations to convert between them.</p>
<p class="MsoNormal"> <span></span></p><a name='more'></a><p></p>
<p class="MsoNormal"><b><span style="font-size: 16pt; line-height: 107%;">Probability
and (Statistical) Odds</span></b></p>
<p class="MsoNormal">Everything comes from probability, so let's define that
first.</p>
<p class="MsoNormal">In a statistical context, probability is the ratio of number of
times that an event does happen to the number of times that it could happen,
under identical conditions.</p>
<p class="MsoNormal">For example, the probability that a six-sided die rolls a
"3" is 1/6 or "1 in 6" because if that die were rolled 6
million (without wearing out), it would roll a "3" about 1 million
times*.</p>
<p class="MsoNormal">As another example, in 2012 a pseudo-Canadian coffee chain
did a promotion where there were 45 million coffee cups with prizes out of 270
million coffee cups in total. The probability of winning a prize was the number
of winning cups divided by the number of total cups. (Assuming that winning
cups were evenly distributed).</p><p> <br /></p><p class="MsoNormal">That means the probability of any cup being a winner was <span style="font-family: "Courier New";">45,000,000 / 270,000,000</span>,</p>
<p class="MsoNormal">which simplifies to <span style="font-family: "Courier New";">45/270</span></p>
<p class="MsoNormal">and again to <span style="font-family: "Courier New";">1/6</span>.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Next, let's define odds in a statistical context</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The odds of an event is the fraction <span style="font-family: "Courier New";">Pr(Event) / Pr(Not Event)</span>,</p>
<p class="MsoNormal">or, equivalently, </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Odds = Pr(Event) /
(1 - Pr(Event)</span>.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">In both the coffee cup and dice examples, <span style="font-family: "Courier New";">Pr(event) = 1/6</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">That means the odds of the event (e.g., rolling a 3, choosing
a winning coffee cup) would be </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">(1/6)<span style="mso-spacerun: yes;"> </span>/<span style="mso-spacerun: yes;"> </span>(1 -
1/6) = </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">(1/6) / (5/6) = </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">(1/5)</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Meaning that the event (e.g. a winning cup) will happen 1
time for every 5 times that it doesn't happen (e.g. a losing cup). There are 5
times as many losing cups as winning cups.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The way to say the odds is of this is "1 to 5" or
"5 to 1 against". [Foot 1]</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">When a term like "5 to 1 odds against" is used in
a gambling context, it can also mean a statement about the prize to be won if
an event happens. Whether it's about probability or about payouts is usually
clear from context.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">For example:</p>
<p class="MsoNormal">"Betting on certain number in roulette pays
35-to-1" means that you'll win 35 times your bet (and your original bet).</p>
<p class="MsoNormal">"Blackjack pays 3-to-2" means you will win $3 for
every $2 bet when you get a blackjack (as well as your $2 back).</p>
<p class="MsoNormal">"The chance of rolling a total of 7 on two 6-sided dice
is 5-to-1 against" means Pr(not 7) / Pr(7) is 5/1.</p>
<p class="MsoNormal">"Funny Cide is 6-to-5 odds to win the horse race"
means you'll win 6/5 or 1.2 times your bet (and your original bet back).</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">For the first three of these statements about payouts, the
paying event is usually a little too rare to make the bet a consistently
profitable one. The opportunities for an 'edge', or profitable advantage are
either tiny or non-existent in roulette and blackjack. Opportunities for an
edge in horse racing (and other live events) exists more often because you're often
betting against other people, not the house. [Foot 2]</p>
<p class="MsoNormal">This isn't the only way the term 'odds' is used.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b><span style="font-size: 16pt; line-height: 107%;">Decimal
odds</span></b></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">"Decimal odds" is a gambling term. Like other
gambling types of odds, the term describes the size of a potential payout, not
the probably of that payout.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Decimal odds can be calculated as:</p>
<p class="MsoNormal"><span style="font-family: "Courier New";">(Possible total
payout) / (Amount bet)</span></p>
<p class="MsoNormal">where the payout includes your original bet.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">For example: The decimal odds being offered on the Vancouver
Canucks to beat the Calgary Flames win are currently <b>2.5</b>.</p>
<p class="MsoNormal">That means spending $1 to bet on the Canucks to win will pay
out $2.50 if they do win (your original $1 plus a gain of $1.50) and pay out $0
if they don't. </p>
<p class="MsoNormal">If you're just betting with your friend and there's nobody
in the middle to take a cut, then from their perspective, they would be betting
$1.50 for a potential payout of $2.50 (Their $1.50 plus a gain of $1.00).</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">From their perspective, the decimal odds of the Calgary Flames
winning are:</p>
<p class="MsoNormal"><span style="font-family: "Courier New";">(Possible payout) /
(Amount bet) = </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">2.50 / 1.50 = 1.67</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Which means your friend is getting a possible payout of <b>$1.67
for every $1 they put at risk</b>.</p>
<p class="MsoNormal">Why would your friend take on such a bet if the odds they
were getting were worse than yours? It might be fair if you both believed Flames
were a better team that night. Maybe they found a hot, new goalie from
somewhere? </p>
<p class="MsoNormal">In short, your friend might be willing to take worse odds on
the bet if the probability of the Flames winning was higher. That brings us to
a closely related term: <b>implied probability</b>. </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b><span style="font-size: 16pt; line-height: 107%;">Implied
probability</span></b></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">"Implied probability" is the probability an event
has to have for a bet to neither winning nor lose money in the long run.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Implied probability can be calculated as <span style="mso-spacerun: yes;"> </span><span style="font-family: "Courier New";">1 /
Decimal Odds</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">For example, a decimal odds offer of 2.50 for the Vancouver
Canucks implies that the Canucks have a <span style="font-family: "Courier New";">1
/ 2.50 = 0.40 </span>or 40% probability to win.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">We can check our work by calculating the expected payout.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Expected payout = </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Pr(Payout) * (Value
of payout) </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">= 0.40 * 2.5 = 1</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">For, here, we can find that the expected profit (also called
the expected value) of the bet is zero, confirming that the bet is fair and neither
bettor has an edge.</p>
<p class="MsoNormal"><span style="font-family: "Courier New";">E[Profit] =
E[Payoff] - E[Cost] = 1 - 1 = 0</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Your friend who bet against you got decimal odds of 1.67 for
betting on the Flames. The implied probability of such odds are:</p>
<p class="MsoNormal"><span style="font-family: "Courier New";">1 / Decimal Odds = 1
/ 1.67 = 0.60 </span><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">or 60%.</span></p>
<p class="MsoNormal"><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><br /></span></p>
<p class="MsoNormal"><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Your friend's expected payout is: </span><span style="font-family: "Courier New";">Pr(Payout) X (Value of payout) = 0.60 * 1.67 = 1 <br /></span></p>
<p class="MsoNormal">That should make sense because then <span style="font-family: "Courier New";">Pr(Canucks win) + Pr(Flames win) = 0.40 + 0.60 = 1</span>. Therefore,
we are certain that one team will win. (Note that in NHL Hockey there are no
draws and no way to both or neither team to win.)</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Realistically, a decimal odds of 2.5 means that the market
maker (e.g. a bookie or a casino) offering the odds thinks the Canucks have a
slightly lower than 40% chance to win, because the market maker setting the
odds typically sets them to make a small profit.</p>
<p class="MsoNormal">If you have a good reason to think that the Canucks have a
higher than 40% chance of winning tonight (e.g. 45%), then it's a good bet to
make, as long as your estimate is accurate.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Implied probability can be calculated from many types of
gambling odds, and we can use it as a method to translate between different odds.
One of these different odds is American odds.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b><span style="font-size: 16pt; line-height: 107%;">American
odds</span></b></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">"American odds" also describe the size of a
payout, but they do in terms of the profit you will make if you win your bet.</p>
<p class="MsoNormal">In our example hockey game, the decimal odds for the Canucks
to win was 2.5, mean a $1 bet would pay $2.50 if the Canucks won. That $2.50
includes the $1 that you spent to enter the bet. In other words, it's a profit
of $1.50 if you win the bet and a loss $1.00 if you lose the bet.</p>
<p class="MsoNormal"><b>American odds describe the net profit.</b></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Here, the American odds being offered on a Canucks win is <b>+150</b>.</p>
<p class="MsoNormal">If the American odds are positive, then they show "how much
I will I profit for betting $1?".</p>
<p class="MsoNormal">Betting on something with +150 American odds means that a
$1.00 bet will produce a $1.50 gain if the Canucks win.</p>
<p class="MsoNormal">If you're just betting with your friend and there's nobody
in the middle to take a cut, then your friend is betting $1.50 to produce a
possible gain of $1.00 if the Flames win.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The American odds being offered on a Flames win is <b>-150</b>.</p>
<p class="MsoNormal">If the American odds are negative, then they show "how
much must I bet to gain $1?".</p>
<p class="MsoNormal">It's a confusing system at first, but unlike other American
measurement systems, this one has some logical value.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">American odds have a useful symmetry. Ignoring the market
maker's cut, if betting for something has American odds of +X then betting
against that same thing has American odds of -X.</p>
<p class="MsoNormal">When using a market maker, the odds will be asymmetic to
reflect the 'cut' (also called the 'vig', 'vigorish', 'rake', or 'juice') that
the market maker takes for their service as a market maker. Instead of
+150/-150, you might see +145/-155 or +140/-160. </p>
<p class="MsoNormal">After including the market maker's cut, the positive side
will always be a smaller number than the negative side. If it wasn't (e.g.
+160/-140), then there would be an arbitrage opportunity where you could bet on
both sides and make a guaranteed profit no matter who won.</p>
<p class="MsoNormal">If the event in question is a 50/50 chance, or is very close
to one, you might even see negative odds on both sides. For example, one thing
people bet on for the NFL super bowl is the coin toss and market makers often
offer odds of -105/-105 in American odds, or about 1.95/1.95 in decimal odds
for either side of the coin.</p>
<p class="MsoNormal">There is no such thing as +50 or -75 in American odds, or
any other value between -99 and +99. Odds of +50 are written as -200. Odds of
-75 are written as +133. Both +100 and -100 refer to a potential net gain of $1
for a bet of $1.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b><span style="font-size: 16pt; line-height: 107%;">Translating
between odds</span></b></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Let A be the American odds, D be the decimal odds, and P be
the implied probability.</p>
<p class="MsoNormal"><br /></p>
<p class="MsoNormal">If American odds are positive, they describe what you get
for $1, not including your $1 back, so </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">If A > 0:<span style="mso-spacerun: yes;"> </span></span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">D = (A / 100) + 1</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">A = (D * 100) - 100</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">P = 100 / (100 + A)</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">If American odds are negative, they describe what it costs
to gain $1, so the reciprocal value is used.</p>
<p class="MsoNormal"><span style="font-family: "Courier New";">If A < 0:</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">D = 1 - 100/A</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">A = 100 / (1-D)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">P = 1 - (100 / (100
- A))</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The R code for all the American-Decimal-Implied Probability
translations is here:</p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"># Note: Enter in ONE of A, D,
or P (American, Decimal, implied Probability)</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"># If you enter in two or
more, and there are contractions</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"># Decimal odds precede
implied probability precede American odds</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; font-size: 9pt; mso-ansi-language: PT-BR;">convert_odds = function(A=NA, D=NA, P=NA)</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;">{</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>if(!is.na(D)){P = 1/D}</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>if(!is.na(P)){D = 1/P}</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"><br /></span></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>if(!is.na(D) & D >= 2)</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>{</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 2;"> </span>A = (D * 100) - 100 </span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>}</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>if(!is.na(D) & D < 2)</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>{</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 2;"> </span>A = 100 / (1-D) </span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>}</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"> </span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>if(!is.na(A) & A > 0)</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>{</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 2;"> </span>D = (A / 100) + 1</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 2;"> </span>P = 100 / (100 + A)</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>}</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 2;"> </span></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>if(!is.na(A) & A < 0)</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>{</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 2;"> </span>D = 1 - 100/A</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 2;"> </span>P = 1 - (100 / (100 - A))</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>}</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;"><span style="mso-tab-count: 1;"> </span>return(list(A=A, D=D, P=P))</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New"; font-size: 9pt;">}</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Finally, the term "long odds" refers to rare
events and the term "short odds" refers to common ones. In the hockey
example, the Canucks have the longer odds because they have a larger payoff and
a smaller implied probability.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Follow-up questions:</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>Q1:</b> Check that all the formulas work by</p>
<p class="MsoNormal">A) Finding A and P when D = 2.5</p>
<p class="MsoNormal">B) Finding D and P when A = -150</p>
<p class="MsoNormal">C) Finding A and D when P = 0.50</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>Q2:</b> Find a betting strategy in which the American
odds of +160 on outcome and -140 on the opposite outcome is 100% guaranteed to
make a profit. (Hint: You need to bet on both sides, but how much?)</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>Q3:</b> The 2021 NFL Super Bowl (also called Super Bowl LV)
offered a bet on whether the game will be a "scorigami", which is a
final game score that has never happened in an NFL game before. Two days before
the event, the odds being offered were +1100 for the game to be a scorigami,
and -1400 to not be a scorigami. </p>
<p class="MsoNormal">A) Find the decimal odds and implied probability of each
side.</p>
<p class="MsoNormal">B) Explain why the sum of the two implied probabilities is
more than 100%. Was there a mistake in the betting line, or did something else
happen?</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>Q4:</b> Make and test your own R functions to translate
from American odds to decimal odds to implied probability.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">[Foot 1] Sometimes advertisers will write "odds are 1
in X" when they mean "probably 1 in X" for some prize.
Technically, they're delivering something better than advertised, so they get
away with it.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">[Foot 2] Edges in roulette do exist from imperfect roulette
wheels, but it's a great deal of work to identify such wheels and maintenance
is regular enough that making a profit this way is nearly impossible, and a
colossal amount of work.<br />
<br />
Card counting is a famous edge in Blackjack, but like roulette, it's a lot of
work for a minor and unreliable gain. Casinos can protect themselves by using
many decks of cards together, and eliminate any accumulated edge at any time by
reshuffling the decks.</p>
<p class="MsoNormal"> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjntOXWX2iqnwGutHXuLTS-NDpGpIQRSWYVqSIpTZ572nEzy4C1FCS4_UBtg0ByOq1QRYpOaEibhP44rERO15pUmSh54XqMghyAAfIp7XeYQHtPhlEIKrGC3Kex3pUNjPTIUsJyTGn2LJrn/s831/Chica+Lake+1+Mud.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="829" data-original-width="831" height="351" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjntOXWX2iqnwGutHXuLTS-NDpGpIQRSWYVqSIpTZ572nEzy4C1FCS4_UBtg0ByOq1QRYpOaEibhP44rERO15pUmSh54XqMghyAAfIp7XeYQHtPhlEIKrGC3Kex3pUNjPTIUsJyTGn2LJrn/w352-h351/Chica+Lake+1+Mud.png" width="352" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Gambling can be an addiction. <br />Know your limit, or there's end to the mess you'll find yourself in.<br /></td></tr></tbody></table><br /></p>
<p class="MsoNormal">To be covered in later notes on odds:</p>
<p class="MsoNormal">Win / Loss / Push (and other multi-way outcomes)</p>
<p class="MsoNormal">Parleys/Multipliers</p>
<p class="MsoNormal">Asian half-odds</p>
<p class="MsoNormal">Prediction market (classic and Twitch)</p>
<p class="MsoNormal">The mechanics of vig (and other places it shows up, like
poker tournaments and perimutual markets)</p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-10991079803082885722021-02-14T13:21:00.006-08:002021-02-16T15:51:09.689-08:00 How does Polychoric Correlation Work? (aka Ordinal-to-Ordinal correlation)Let's say you've got data of many
paired cases of two ordinal variables, like you might when you ask a large
number of people the same two Likert scale questions (e.g. "poor",
"fair", "good", "very good",
"excellent").
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">What could you learn from the data
from those two questions?</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Here's a few common approaches:</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">1) Compare the means of each
variable by abusing a t-test.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">2) Compare the distribution of
each variable with a chi-squared goodness-of-fit test.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">3) Check for a relationship
between responses of each variable with a chi-squared independence test. </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">4) Estimate the strength of such a
relationship with a Spearman correlation.</p><p></p><span><a name='more'></a></span><p><br /></p><p class="MsoNormal" style="margin-bottom: 0cm;">Spearman correlation works because
it's non-parametric; it doesn't care about the distribution of the variables, but
it leaves a lot of information unused. There's a better alternative for
ordinal-to-ordinal data: Polychoric correlation.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Polychoric correlation is designed
for to find correlations between ordinal variables (ordered categorical data)
like Likert scales and binned data.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-size: 18pt; line-height: 107%;">Great, but how does it work?</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Polychoric correlation assumes
that each ordinal data point represents a binned continuous value from a normal
distribution, and then tries to estimate the correlation coefficient on that
assumption.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">For actual binned data like responses
to "How many movies did you watch this year: 0-4, 5-9, 10-19, 20 or more?",
then that's nothing new to say that a "10-19" response could
represent someone who has watched 12 movies.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">With Likert data like responses to
"How was your last movie: poor, fair, good, very good, or excellent?",
we treat the responses like they're binned values for some abstract variable we
didn't ask for, like a quality scale from 0-100, so "very good" could
be a 73 or a 64. </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The actual number of movies
watched, or the actual 0-100 quality score that someone would give their last
movie aren't recorded in the data, so we would call them <b><i>latent </i></b>variables.
</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">In polychoric correlation, we don't
need to know or specify where the boundary between "good" and
"very good" is, just that it exists. The distribution of the ordinal
responses, along with the assumption that the latent values follow a normal
distribution, is enough that the <span style="font-family: "Courier New";">polychor()</span>
function in the <span style="font-family: "Courier New";">polycor</span> R
package can do that for us. In most practical cases, you don't even need to
know where the cutoffs are, but they are useful for demonstration that the method
works.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Polychoric correlation estimates
the correlation between such latent variables as if you actually knew what
those values were. In the examples given, we start with the latent variables
and use cutoffs to set them into bins, and then use polychoric on the artificially
binned data. In any practical use case, the latent data would be invisible to
you, and the cutoffs would be determined by whoever designed the survey.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The only reason the latent
variables are visible in these examples is to demonstrate how effective polychoric
correlation is under different conditions. Ideally, polychoric correlation on
the (realistic) binned / ordinal data will closely match the Pearson
correlation on the latent data.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">To start, here's the modified example
code given in the help file for <span style="font-family: "Courier New";">polychor()</span>.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">This code first checks that you
have <span style="font-family: "Courier New";">mvtnorm</span> installed, which is
used to generate 1000 values from a standard multivariate normal distribution with
covariance 0.5.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">library(polycor)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">library(mvtnorm)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">data
<- rmvnorm(1000, c(0, 0), matrix(c(1, .5, .5, 1), 2, 2))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Then it takes the Pearson correlation of those data points.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(data)
# 0.5264</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
And the Spearman correlation<span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(data,
method="spearman") #0.5043</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Now let's try with polychoric correlation.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
First we need some cutoffs to break the data into cells</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
ideally with breakpoints that avoid nearly empty cells.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x
<- cut(dat[,1], c(-Inf, .75, Inf))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y
<- cut(dat[,2], c(-Inf, -1, .5, 1.5, Inf))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Then take the polychoric correlation.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
The default method uses the faster 2-step method and only returns the
correlation.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">polychor(x,
y)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] 0.5230474</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><br /></p><p class="MsoNormal" style="margin-bottom: 0cm;">Setting standard error to <span style="font-family: "Courier New";">TRUE</span> gives us a standard error of the
correlation and a chi-squared test for bivariate normality. If the data is
approximately normal, the normality test should yield a moderate p-value
(between .025 and .975).</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">polychor(x,
y, std.err=TRUE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Polychoric Correlation, 2-step est. = 0.5231 (0.03729)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Test of bivariate normality: Chisquare = 2.758, df = 2, p = 0.2519</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Setting ML to <span style="font-family: "Courier New";">TRUE</span> changes the method the slower
Maximum Likelihood method but it also allows for estimates of the cutoffs
between cells of the latent, unseen normally distributed values that have been
binned into cells. Compare the estimates to the true cutoffs of .75 and
c(-1.00, 0.50, 1.50)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">polychor(x,
y, ML=TRUE, std.err=TRUE)<span style="mso-spacerun: yes;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Polychoric
Correlation, ML est. = 0.5231 (0.03819)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Test
of bivariate normality: Chisquare = 2.739, df = 2, p = 0.2543</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Row Threshold</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Threshold Std.Err.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>0.7537<span style="mso-spacerun: yes;">
</span>0.04403</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Column Thresholds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Threshold Std.Err.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#1<span style="mso-spacerun: yes;"> </span>-0.9842<span style="mso-spacerun: yes;">
</span>0.04746</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#2<span style="mso-spacerun: yes;"> </span>0.4841<span style="mso-spacerun: yes;">
</span>0.04127</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#3<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>1.5010<span style="mso-spacerun: yes;">
</span>0.06118</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Let's plot this out with dividing lines</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">plot(data)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">abline(h=c(-1,
.5, 1.5))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">abline(v=c(-.75))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p><p class="MsoNormal" style="margin-bottom: 0cm;"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPwonMp9Lovtqg8X5GebXXORIceig1GbRIdZ9n2Vkl66Ok52VGuTVvdldWn3-G2kNvxPZMeF660wSFQkAxMJ_NxGrVfuKKbMK12OT3D5Uj55WWBkw5_1JVNzFLCREs1fCGNt_n6pQuPgAm/s907/Polychor+Fig+1.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="747" data-original-width="907" height="330" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPwonMp9Lovtqg8X5GebXXORIceig1GbRIdZ9n2Vkl66Ok52VGuTVvdldWn3-G2kNvxPZMeF660wSFQkAxMJ_NxGrVfuKKbMK12OT3D5Uj55WWBkw5_1JVNzFLCREs1fCGNt_n6pQuPgAm/w400-h330/Polychor+Fig+1.png" width="400" /></a></div><br /><span style="font-family: "Courier New";"><br /></span><p></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New"; mso-no-proof: yes;">
</span><span style="font-family: "Courier New";"></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">You don't need all the
multivariate random normal overhead, <span style="font-family: "Courier New";">polychor()</span>
works on any crosstab. We can check that this with the dataset we already made.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">table(x,y)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody><tr style="height: 17.95pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid windowtext; height: 17.95pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 33.7pt;" valign="top" width="45">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">155</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 17.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 33.7pt;" valign="top" width="45">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">437</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 17.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 33.7pt;" valign="top" width="45">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">162</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 17.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 33.7pt;" valign="top" width="45">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">21</span></p>
</td>
</tr>
<tr style="height: 17.95pt; mso-yfti-irow: 1; mso-yfti-lastrow: yes;">
<td style="border-top: none; border: 1pt solid windowtext; height: 17.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 33.7pt;" valign="top" width="45">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">7</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 17.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 33.7pt;" valign="top" width="45">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">88</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 17.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 33.7pt;" valign="top" width="45">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">84</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 17.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 33.7pt;" valign="top" width="45">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">46</span></p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">polychor(tab)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
0.5230474</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Let's try the same data with some
different cutoffs and see how it does</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x2
<- cut(data[,1], c(-Inf, -2, -1, 0, 1, 2, Inf))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y2
<- cut(data[,2], c(-Inf, -2, -1, 0, 1, 2, Inf))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">table(x2,y2)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody><tr style="height: 22pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid windowtext; height: 22pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">4</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">7</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">15</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">1</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
</tr>
<tr style="height: 20.75pt; mso-yfti-irow: 1;">
<td style="border-top: none; border: 1pt solid windowtext; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">10</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">38</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">49</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">25</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">5</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
</tr>
<tr style="height: 22pt; mso-yfti-irow: 2;">
<td style="border-top: none; border: 1pt solid windowtext; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">7</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">58</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">147</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">102</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">28</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">1</span></p>
</td>
</tr>
<tr style="height: 20.75pt; mso-yfti-irow: 3;">
<td style="border-top: none; border: 1pt solid windowtext; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">2</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">31</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">92</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">158</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">50</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">2</span></p>
</td>
</tr>
<tr style="height: 22pt; mso-yfti-irow: 4;">
<td style="border-top: none; border: 1pt solid windowtext; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">5</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">22</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">62</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">39</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">12</span></p>
</td>
</tr>
<tr style="height: 20.75pt; mso-yfti-irow: 5; mso-yfti-lastrow: yes;">
<td style="border-top: none; border: 1pt solid windowtext; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">3</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">8</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">11</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 20.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 36.2pt;" valign="top" width="48">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">6</span></p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">plot(data)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">abline(h=c(
-2, -1, 0, 1, 2))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">abline(v=c(
-2, -1, 0, 1, 2))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhV7fvmARXaTapJ5pxs3Hlj3RMVcGe0_d8TtqP59Z2MAeO4My7W_dfh1fnD6Ogdm2-dUszTz_z_2u9dqX_B-ddZv4NSytJom42k7viYO7BP5KQfNWGDTLZAJLDA_1GAtUstBMgFs1561Y6Q/s907/Polychor+Fig+2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="747" data-original-width="907" height="330" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhV7fvmARXaTapJ5pxs3Hlj3RMVcGe0_d8TtqP59Z2MAeO4My7W_dfh1fnD6Ogdm2-dUszTz_z_2u9dqX_B-ddZv4NSytJom42k7viYO7BP5KQfNWGDTLZAJLDA_1GAtUstBMgFs1561Y6Q/w400-h330/Polychor+Fig+2.png" width="400" /></a></div><br /><p></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New"; mso-no-proof: yes;">
</span><span style="font-family: "Courier New";"></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">polychor(x2,y2,ML=TRUE,std.err=TRUE)
# takes 5-10 seconds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Polychoric
Correlation, ML est. = 0.5069 (0.02623)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Test
of bivariate normality: Chisquare = 22.89, df = 24, p = 0.5266</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Row Thresholds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Threshold Std.Err.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#1
-1.915000<span style="mso-spacerun: yes;"> </span>0.08103</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#2
-1.016000<span style="mso-spacerun: yes;"> </span>0.04787</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#3
-0.009191<span style="mso-spacerun: yes;"> </span>0.03940</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#4<span style="mso-spacerun: yes;"> </span>0.962900<span style="mso-spacerun: yes;">
</span>0.04710</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#5<span style="mso-spacerun: yes;"> </span>1.928000<span style="mso-spacerun: yes;">
</span>0.08108</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Column Thresholds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Threshold Std.Err.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#1<span style="mso-spacerun: yes;"> </span>-1.97800<span style="mso-spacerun: yes;">
</span>0.08591</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#2<span style="mso-spacerun: yes;"> </span>-0.98300<span style="mso-spacerun: yes;">
</span>0.04712</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#3<span style="mso-spacerun: yes;"> </span>-0.02738<span style="mso-spacerun: yes;">
</span>0.03947</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#4<span style="mso-spacerun: yes;"> </span>1.01800<span style="mso-spacerun: yes;">
</span>0.04797</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#5<span style="mso-spacerun: yes;"> </span>2.05600<span style="mso-spacerun: yes;">
</span>0.09097</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The <span style="font-family: "Courier New";">polychor()</span>
function estimated the thresholds well, but with a little less accuracy and
certainty at the edges than the middle. It estimated the correlation near the
true value of 0.5 pretty well. Finally, the chi-squared test yielded a moderate
p-value, indicating that there wasn't any significant difference between our
binned data and typical data binned from a normal distribution. In short, it
worked great.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Finally, let's try it with some
non-linear data, but with evenly-spaced cutoffs.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Make random data in a wide s-curve and force it to mean 0, sd 1.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x3_contin
= runif(1000)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y3_contin
= 4*(x3_contin -0.5)^3 + runif(1000) - 0.5</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x3_contin
= (x3_contin - mean(x3_contin)) / sd(x3_contin)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y3_contin
= (y3_contin - mean(y3_contin)) / sd(y3_contin)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">plot(x3_contin,y3_contin)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">abline(h=c(-1,1))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">abline(v=c(-0.5,0.5))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjCsylj8UJvVipeIUt4EnOsamDlWhWXCjyP1vDLoGnqqaWJkLPgKbf1KSArQ-t5NmdMhDtcqU4jGiy-xfM-EnqoYZYpNEiH9edLmhzYGvevGWJhf5OGWrnpmvNGIyfef_PyzrGKJ-c4s-_/s672/Polychor+Fig+3.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="671" data-original-width="672" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjCsylj8UJvVipeIUt4EnOsamDlWhWXCjyP1vDLoGnqqaWJkLPgKbf1KSArQ-t5NmdMhDtcqU4jGiy-xfM-EnqoYZYpNEiH9edLmhzYGvevGWJhf5OGWrnpmvNGIyfef_PyzrGKJ-c4s-_/s320/Polychor+Fig+3.png" width="320" /></a></div><br /> <p></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Pearson</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(x3_contin,y3_contin)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] 0.5021618</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Spearman </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(x3_contin,y3_contin,method="spearman")</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#[1]
0.4696544</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Polychoric</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x3_ord
= cut(x3_contin, c(-Inf, -0.5, 0.5, Inf))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y3_ord
= cut(y3_contin, c(-Inf, -1, 1, Inf))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">table(x3_ord,
y3_ord)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody><tr style="height: 23.55pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid windowtext; height: 23.55pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.15pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">121</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 23.55pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.2pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">211</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; height: 23.55pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.2pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">16</span></p>
</td>
</tr>
<tr style="height: 22.2pt; mso-yfti-irow: 1;">
<td style="border-top: none; border: 1pt solid windowtext; height: 22.2pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.15pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">52</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22.2pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.2pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">208</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22.2pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.2pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">28</span></p>
</td>
</tr>
<tr style="height: 22.2pt; mso-yfti-irow: 2; mso-yfti-lastrow: yes;">
<td style="border-top: none; border: 1pt solid windowtext; height: 22.2pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.15pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">21</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22.2pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.2pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">211</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; height: 22.2pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 49.2pt;" valign="top" width="66">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">132</span></p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">polychor(x3_ord,y3_ord,ML=TRUE,std.err=TRUE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Polychoric
Correlation, ML est. = 0.5429 (0.03126)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Test
of bivariate normality: Chisquare = 6.296, df = 3, p = 0.09808</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Row Thresholds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Threshold Std.Err.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#1<span style="mso-spacerun: yes;"> </span>-0.3957<span style="mso-spacerun: yes;">
</span>0.04054</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#2<span style="mso-spacerun: yes;"> </span>0.3415<span style="mso-spacerun: yes;">
</span>0.04056</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Column Thresholds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Threshold Std.Err.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#1<span style="mso-spacerun: yes;"> </span>-0.8624<span style="mso-spacerun: yes;">
</span>0.04531</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#2<span style="mso-spacerun: yes;"> </span>0.9315<span style="mso-spacerun: yes;">
</span>0.04675</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">##
Compare to Pearson, Spearman correlation on binned data</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(as.numeric(x3_ord),
as.numeric(y3_ord)) #</span> <span style="font-family: "Courier New";">0.4217</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(as.numeric(x3_ord),
as.numeric(y3_ord), method="spearman") #</span> <span style="font-family: "Courier New";">0.4217</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The correlation coefficients on the
latent, unbinned data are 0.4696 and 0.5021. That's a good, informal target
range for the 'true' correlation. Here, </span><span style="font-family: "Courier New";">polychor()</span><span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">
overestimated the correlation as 0.5429, but it did a little better than the classic
correlation measures on the binned data, which underestimated the correlation
as 0.4217.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><br /></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The cutoffs in both dimensions
were too close to the middle, but only by 2.5 standard errors, so it's not
egregiously wrong. The polychoric correlation is close the other correlation
measures on the latent data, not the test of bivariate normality failed to
detect that the data was non-normal.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Let's try one more example with
exponential bins.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">set.seed(12345)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x4_contin
= rexp(1000)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y4_contin
= x4_contin * exp(runif(1000, min=log(0.5), max=log(2))) + 3*runif(1000)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x4_contin
= (x4_contin - mean(x4_contin)) / sd(x4_contin)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y4_contin
= (y4_contin - mean(y4_contin)) / sd(y4_contin)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x_cut
= min(x4_contin) + c(0.5,1,2,4)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y_cut
= min(y4_contin) + c(0.5,1,2,4)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">round(x_cut,2)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] -0.48<span style="mso-spacerun: yes;"> </span>0.02<span style="mso-spacerun: yes;"> </span>1.02<span style="mso-spacerun: yes;">
</span>3.02</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">round(y_cut,2)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
[1] -1.03 -0.53<span style="mso-spacerun: yes;"> </span>0.47<span style="mso-spacerun: yes;"> </span>2.47</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">plot(x4_contin,
y4_contin)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">abline(v=x_cut)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">abline(h=y_cut)</span></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuQ5RkFArLRbu8M4Z1zAIYjLHmwAmHtvrqC0cA3JdZ0CiJqoRp2wJww5MbtZ3bYKhYLfkR-QQ10wnOlWp7l4bEMt99lepjnrxmJ37wFSFkxkZwe-_ugsi0zOyVWhjSrQ0mmbEz20QmTIst/s672/Polychor+Fig+4.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="671" data-original-width="672" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuQ5RkFArLRbu8M4Z1zAIYjLHmwAmHtvrqC0cA3JdZ0CiJqoRp2wJww5MbtZ3bYKhYLfkR-QQ10wnOlWp7l4bEMt99lepjnrxmJ37wFSFkxkZwe-_ugsi0zOyVWhjSrQ0mmbEz20QmTIst/s320/Polychor+Fig+4.png" width="320" /></a></div><p></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New"; mso-no-proof: yes;">
</span><span style="font-family: "Courier New";"></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Latent correlations</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(x4_contin,y4_contin)
# 0.7152</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(x4_contin,y4_contin,method="spearman")
# 0.5781 </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Polychoric</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">x4_ord
= cut(x4_contin, c(-Inf, x_cut, Inf))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">y4_ord
= cut(y4_contin, c(-Inf, y_cut, Inf))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">table(x4_ord,
y4_ord)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">99</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">105</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">183</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">11</span></p>
</td>
<td style="border-left: none; border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border-top: none; border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">21</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">56</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">123</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">45</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border-top: none; border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">3</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">33</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">104</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">85</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">1</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="border-top: none; border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">2</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">33</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">59</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">20</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 4; mso-yfti-lastrow: yes;">
<td style="border-top: none; border: 1pt solid windowtext; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">0</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">8</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-color: currentcolor windowtext windowtext currentcolor; border-left: none; border-right: solid windowtext 1.0pt; border-style: none solid solid none; border-top: none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 93.5pt;" valign="top" width="125">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;"><span style="font-family: "Courier New";">9</span></p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">polychor(x4_ord,y4_ord,ML=TRUE,std.err=TRUE)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Polychoric
Correlation, ML est. = 0.6688 (0.02182)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Test
of bivariate normality: Chisquare = 48.03, df = 15, p = 2.519e-05</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Row Thresholds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Threshold Std.Err.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#1<span style="mso-spacerun: yes;"> </span>-0.2707<span style="mso-spacerun: yes;">
</span>0.03957</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#2<span style="mso-spacerun: yes;"> </span>0.3434<span style="mso-spacerun: yes;">
</span>0.04026</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#3<span style="mso-spacerun: yes;"> </span>1.1220<span style="mso-spacerun: yes;">
</span>0.05072</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#4<span style="mso-spacerun: yes;"> </span>2.1960<span style="mso-spacerun: yes;">
</span>0.09929</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Column Thresholds</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>Threshold Std.Err.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#1<span style="mso-spacerun: yes;"> </span>-1.1360<span style="mso-spacerun: yes;">
</span>0.05041</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#2<span style="mso-spacerun: yes;"> </span>-0.4750<span style="mso-spacerun: yes;">
</span>0.04056</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#3<span style="mso-spacerun: yes;"> </span>0.6973<span style="mso-spacerun: yes;">
</span>0.04375</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#4<span style="mso-spacerun: yes;"> </span>1.9430<span style="mso-spacerun: yes;">
</span>0.08171</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">##
Compare to Pearson, Spearman correlation on binned data</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(as.numeric(x4_ord),
as.numeric(y4_ord)) #</span> <span style="font-family: "Courier New";">0.5747</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">cor(as.numeric(x4_ord),
as.numeric(y4_ord), method="spearman") #</span> <span style="font-family: "Courier New";">0.5600</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">This was by far the most challenging
case for the <span style="font-family: courier;">polychor()</span> function because a row and column are nearly empty, and
several cells are completely empty. Some of the cutoff estimates are off, especially
in the tail where there are fewer data points. The test of bivariate normality has
also detected the non-normality. </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p><p class="MsoNormal" style="margin-bottom: 0cm;">However what matters most is that
the estimated correlation coefficient is a reasonable number, and it's near the
middle of our target range of 0.5781 to 0.7182, which we got from the unbinned correlations.
By comparison, classic correlation methods missed the range by a little bit. that's
the measurement that matters. So ultimately, <span style="font-family: courier;">polychor()</span> did the job, even in far
from ideal conditions, so it's worth considering the next time you're dealing
with ordinal data.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">If you liked this and you deal
with ordinal data a lot, see my previous post on 'debinning', or estimating a continuous
latent distribution from binned data at <a href="https://www.stats-et-al.com/2018/10/parameter-estimation-of-binned-data.html">https://www.stats-et-al.com/2018/10/parameter-estimation-of-binned-data.html</a>
.</p><p class="MsoNormal" style="margin-bottom: 0cm;"></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span>The work in this post is based on work done at <a href="https://www.bigriveranalytics.com/">Big River Analytics</a>, an Indigenous-owned data analytics and economics consulting firm based in Terrace, British Columbia, Canada.</span></p>Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-39510919252216761482021-01-30T14:27:00.001-08:002021-01-30T14:32:04.247-08:00Lottery tickets, baseball cards, and the coupon collector's problem
<p class="MsoNormal">A man, a woman, an enby, and 27 elephants walk into a bar.
The bartender looks at the group of 30 and asks "What is the probability
that 2 or more of you have the same birthday?".
</p><p class="MsoNormal"><br /></p><p class="MsoNormal">The group proceeds to disregard twins, leap years, and
building codes. They talk amongst themselves, leave a little space for the
reader to calculate or guess for themselves, </p>
<span><a name='more'></a></span><p class="MsoNormal"><br /></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"># Hint:</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">1 - prod( (365:336)
/ 365)</span></p><p> </p><p> </p><p class="MsoNormal">and respond "70.63%". Is this surprising, given
there are only 30 of them and there are 365 days?</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">That's the Birthday Paradox ( <a href="https://en.wikipedia.org/wiki/Birthday_problem">https://en.wikipedia.org/wiki/Birthday_problem</a>
) - only 23 people are needed for you to have about a 50/50 chance at a
'double' in birthdays. If you flip the problem upside down and ask "how
many people do you need in a group until at least one person had every
birthday", then you have the Coupon Collector's Problem. </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Rather then worry about whether elephants are people, let's
talk about the coupon collecting the way people at Topps and Panini do: with
collectible cards and stickers.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">If there are 365 different cards in a set, how many randomly
selected cards do you need to get one of every card in the set, on average?
Assume that every card is equally likely, and that there are thousands of
copies of each card in the set.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">It takes 1 card to get the first one.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The next card has a 364/365 chance of being a new one. By
the geometric distribution, that means it will take 365/364 cards on average to
get a new one.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">If we have two different cards, the next one has a 363/365
chance of being a new one. Or, 365/363 cards on average to get a new one.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">If we have N different cards, the next one has has a (365 -
N) / 365 chance of being a new one. Or 365 / (365 - N) cards on average to get
a new one.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">When we have all but one card, this gets to be a real pain.
Each card only has a 1/365 chance to be the one we want, so we need 365 cards
on average to find it.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">And how many do we need from the start? Just sum up all
E[cards] to get new ones.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">(365 / 365) + (365 / 364) + (365/363) + ... + (365/3) +
(365/2) + (365/1) </p>
<p class="MsoNormal">=</p>
<p class="MsoNormal">1 + 1.003 + 1.006 + ... + 122.7 + 182.5 + 365</p>
<p class="MsoNormal">=</p>
<p class="MsoNormal">2364.65 cards, or about 6.5 times the number of different
cards in the set.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">You can check the answer yourself in R with the code</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">sum( 365 / (365:1) )
</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">------------</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">So that's the basic Coupon Collector's problem, but there's
lots of way you can mess with it.</p>
<p class="MsoNormal">- How many cards until I get 50% / 75% / 90% of the
collection? (and I trade or buy the rest individually)</p>
<p class="MsoNormal">- What if some cards are rarer?</p>
<p class="MsoNormal">- What if the cards come in packs that guarantee they don't
have doubles?</p>
<p class="MsoNormal">- What if you want 4 copies of each card?</p>
<p class="MsoNormal">- Are sports cards a good investment?</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Instead of birthdays, let's use a real card set for the rest
of this article: 2020 Topps Baseball Update.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">---------------</p>
<p class="MsoNormal"><span style="font-size: 18pt; line-height: 107%;">What if you
want 2 or more copies of each card?</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">With multiple copies, things get much more complex because
of the number of steps involved.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">With 1 copy, the steps are linear, as in this figure, so we
can get an expected number of cards needed get a 1 copy of each card just by
adding up the number of cards to move through each step.</p><p class="MsoNormal"> </p><p class="MsoNormal"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjiGXCM91_mBJAaVN0k7EsYjp9tnRMkiSvzlgqqfRlN8U3qdKzFyZRWQzMAbQ_DW-m4fi9aXAsR5-dSEfJabVDYFOWrAJTaOWQr7AxxaBtyRdNm-KebgT61qiL73HBV1I4BhIcIjgpZf-q/s2442/Coupon+01.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="456" data-original-width="2442" height="94" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjiGXCM91_mBJAaVN0k7EsYjp9tnRMkiSvzlgqqfRlN8U3qdKzFyZRWQzMAbQ_DW-m4fi9aXAsR5-dSEfJabVDYFOWrAJTaOWQr7AxxaBtyRdNm-KebgT61qiL73HBV1I4BhIcIjgpZf-q/w501-h94/Coupon+01.png" width="501" /></a></div><br /><br /><p></p><p class="MsoNormal">But when we need even 2 copies of each card in a set, things
get more complex. There's more than one path to get from (0,0) for 0 first
copies and 0 second copies to (4,4) which is 4 first and second copies, and
computing the expected number of cards to get there becomes complex.</p><p class="MsoNormal"> </p><p class="MsoNormal"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYcYP-WLPyogsG31MGVxO84h0Fy7CKgFa7aKqRkep-DYTCtIoNPbaVQCRetOaMEDNKLUkqwo_3ryn1CMoCid9gatLbDXKRdpQ49KkpKX0EHTL_hTqXQAJ4-WqoMCeWMun0GxPZgMOGFp4Z/s2048/Coupon+02.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1772" data-original-width="2048" height="339" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYcYP-WLPyogsG31MGVxO84h0Fy7CKgFa7aKqRkep-DYTCtIoNPbaVQCRetOaMEDNKLUkqwo_3ryn1CMoCid9gatLbDXKRdpQ49KkpKX0EHTL_hTqXQAJ4-WqoMCeWMun0GxPZgMOGFp4Z/w391-h339/Coupon+02.png" width="391" /></a></div><br /><p></p><p class="MsoNormal"> </p><p class="MsoNormal"><!--[if gte mso 9]><xml>
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<![endif]--></p><p class="MsoNormal">
</p><p class="MsoNormal">We can, however, run a simulator many times to estimate the
distribution of cards needed and take expected value, median, 90th percentile,
or anything else we need from that.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">This simulator code designed to be illustrative, not
efficient. In this example, we are looking for at least 4 copies of each card
from a set where everything is equally likely from a set of 100 cards.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">setsize = 100</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Ncopies_wanted = 4</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">set.seed(12345) #
Remove this line to get a different result when you repeat</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">cardcount = rep(0,
setsize)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">while(
min(cardcount) < Ncopies_wanted)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>newcard = sample(1:setsize, 1)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>cardcount[newcard] = cardcount[newcard] + 1</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">sum(cardcount)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"># 921 cards</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">In one run of the simulation, it took 921 random cards to
get 4 complete sets. How is that compared to getting 1 copy?</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">sum( setsize /
(setsize:1))</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"># 518.73</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">921 / 518.73 = 1.775</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">It took less than twice as long to get four complete sets as
it took to get one complete set. Is that lucky? It's a lot less than 4. If we
run this many times, we can get a better picture.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">cards_to_complete_N_sets
= function(setsize, Ncopies_wanted)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>cardcount = rep(0, setsize)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>while( min(cardcount) < Ncopies_wanted)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 2;"> </span>newcard = sample(1:setsize, 1)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 2;"> </span>cardcount[newcard] =
cardcount[newcard] + 1</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>return( sum(cardcount) )</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Nruns = 1000</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">cards_needed =
rep(NA,Nruns)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">for(k in 1:Nruns)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>cards_needed[k] =
cards_to_complete_N_sets(100, 4)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">mean(cards_needed)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"># 1081.9</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">## Standard error of
the mean</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">sd(cards_needed) /
sqrt(length(cards_needed))</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">hist(cards_needed, n=20,
las=1, col="LightBlue")</span></p>
<p class="MsoNormal"> </p><p class="MsoNormal"></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsQln0omnMjLu9lP6fCXFxz0Xpg0U1ZV1P2t946cajWFjJ6OkJM8tfmtrlR7ODjRHqVX19Q6UKjzLfxwWQanVVpsuYVuK5NkIkiQGHdw85kzKZcUi65nlzjoiGMFc3leF0kghnfwZVe1yG/s985/Coupon+03.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="855" data-original-width="985" height="357" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsQln0omnMjLu9lP6fCXFxz0Xpg0U1ZV1P2t946cajWFjJ6OkJM8tfmtrlR7ODjRHqVX19Q6UKjzLfxwWQanVVpsuYVuK5NkIkiQGHdw85kzKZcUi65nlzjoiGMFc3leF0kghnfwZVe1yG/w410-h357/Coupon+03.png" width="410" /></a></div><br /> <p></p><p class="MsoNormal">According to our figure, 921 was lucky but not unusual. It
takes about 1082 cards on average to complete 4 sets of 100 cards each. That's
still only 2.09 times as many cards as a single set takes. </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">That the number of cards needed for 4 sets (1082) is less
than 4 times the cards needed for 1 set (518.73) is intuitive because you are
farther along in your collection before new random cards start getting wasted.
In other terms, you can start collecting your second copy of each card before
you finish collecting your first copy.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">There's a definite savings in finishing a card set by doing
multiple sets at once.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">How much of a savings? Let's find out. The following code
fills a table of mean cards needed by number of copies and set size.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">set_sizes =
c(50,100,150,200)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">copies =
c(1,2,3,4,5,6,7,8,9,10)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Nruns = 1000</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">mean_cards_needed =
matrix(NA, nrow=length(copies), ncol=length(set_sizes))</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">set.seed(12345)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">for(i in
1:length(copies))</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">for(j in
1:length(set_sizes))</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>cards_needed = rep(NA,Nruns)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>for(k in 1:Nruns)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 2;"> </span>cards_needed[k] =
cards_to_complete_N_sets(set_sizes[j], copies[i])</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>mean_cards_needed[i,j] = mean(cards_needed)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>print(c(i,j,mean_cards_needed[i,j]))</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">round(mean_cards_needed)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Table:</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">>
round(mean_cards_needed)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3] [,4]</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[1,]<span style="mso-spacerun: yes;">
</span>223<span style="mso-spacerun: yes;"> </span>512<span style="mso-spacerun: yes;"> </span>841 1173</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[2,]<span style="mso-spacerun: yes;">
</span>326<span style="mso-spacerun: yes;"> </span>722 1149 1606</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[3,]<span style="mso-spacerun: yes;">
</span>412<span style="mso-spacerun: yes;"> </span>907 1443 2006</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[4,]<span style="mso-spacerun: yes;">
</span>493 1077 1695 2336</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[5,]<span style="mso-spacerun: yes;">
</span>570 1251 1947 2670</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[6,]<span style="mso-spacerun: yes;">
</span>645 1391 2199 2984</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[7,]<span style="mso-spacerun: yes;">
</span>715 1533 2413 3313</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[8,]<span style="mso-spacerun: yes;">
</span>786 1690 2641 3619</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-spacerun: yes;"> </span>[9,]<span style="mso-spacerun: yes;">
</span>852 1841 2850 3917</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">[10,]<span style="mso-spacerun: yes;"> </span>925 1972 3077 4185</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">diff(mean_cards_needed[,1])</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">diff(mean_cards_needed[,2])</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">diff(mean_cards_needed[,3])</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">diff(mean_cards_needed[,4])</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Every additional set takes fewer and fewer additional cards
to complete. </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">------------------------</p>
<p class="MsoNormal"><span style="font-size: 16pt; line-height: 107%;">What if some
cards are rarer? </span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Because each random card has less likelihood of being a rare
card, rare cards are more likely to be the last card needed as well as taking
longer to collect.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Let's consider a set of 11 cards with 10 very common (.0999
chance) and 1 very rare (.001 chance). We can do this my modifying our
simulation function to account for uneven chances.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";">cards_to_complete_N_sets_uneven
= function(card_chance, Ncopies_wanted)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>setsize = length(card_chance)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>cardcount = rep(0, setsize)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>while( min(cardcount) < Ncopies_wanted)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 2;"> </span>newcard = sample(1:setsize, size=1,
prob=card_chance)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 2;"> </span>cardcount[newcard] =
cardcount[newcard] + 1</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>return( sum(cardcount) )</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Nruns = 1000</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">cards_needed =
rep(NA,Nruns)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">card_chances =
c(.001, rep(.0999,10))</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">for(k in 1:Nruns)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>cards_needed[k] =
cards_to_complete_N_sets_uneven(card_chances, 1)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">mean(cards_needed)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"># 1032.449</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">sum( 11 / (11:1))</p>
<p class="MsoNormal"># 33.2</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">If all the cards are equally common, it only takes 33.2
cards to complete the set. But, when one of them is much rarer, the set becomes
MUCH more difficult to complete.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">If the rare card is the last one, then it would take 1 /
.001 or 1000 cards on average to get it and complete the set, so it makes sense
that it would take about 1000 more cards on average to complete a set with even
a single rare one.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">-----------------------------</p>
<p class="MsoNormal"><span style="font-size: 18pt; line-height: 107%;">What if
there are no doubles in a pack of cards?</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">A box of Topps 2020 Update Baseball cards has 60 cards from
their basic set of 300. All 60 of the basic cards are unique. How much does
this change the equation? We can modify our simulator to 'sample without
replacement' to find out.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">packs_to_complete_N_sets
= function(setsize, packsize, Ncopies_wanted)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>cardcount = rep(0, setsize)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>while( min(cardcount) < Ncopies_wanted)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 2;"> </span>newcard = sample(1:setsize,
size=packsize, replace=FALSE)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 2;"> </span>cardcount[newcard] =
cardcount[newcard] + 1</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>return( sum(cardcount) )</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">Nruns = 1000</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">packsize = 60</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">setsize = 300</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">cards_needed =
rep(NA,Nruns)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">set.seed(12345)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">for(k in 1:Nruns)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>cards_needed[k] =
packs_to_complete_N_sets(setsize, packsize, 1)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">mean(cards_needed)</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"># 1736.76</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">mean(cards_needed) /
packsize # packs needed</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>28.946</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";">sum(300 / (300:1)) #
without duplicates</span></p>
<p class="MsoNormal"><span style="font-family: "Courier New";"># 1884.799</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">There's two offsetting factors: A pack with all unique cards
'wastes' fewer cards on its own, but you need to buy the entire pack. Here with
each pack covering 1/5 of the set (60 of 300), it's a net savings in cards.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">------------------</p>
<p class="MsoNormal"><span style="font-size: 18pt; line-height: 107%;">Are sports
cards a good investment? </span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Are sports cards a good investment? That's a matter of
opinion. Here's my (very much non-expert) opinion on the matter: </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The collector value of individual cards, let alone all of
them, is hard to estimate. The value of cards goes up and down based on how
famous a player is and how many people are actively collecting cards as a
whole. That long-term bet is the reason that people make a big deal about
rookie cards of players in their first year - because those players are
relatively unknown and so there's probably fewer copies of a player out there
as a rookie compared to later if they become a superstar.</p>
<p class="MsoNormal">But, given that some people regularly buy up a store's
entire stock of a set to 'flip' the sales later, there's probably some profit
in it with some expertise. If you're looking to invest in sports cards because
you've read recently that it's a hot investment, remember that lots of other
people read those articles too, just like they did in the beanie baby crash.</p>
<p class="MsoNormal">You can't just take those cards are turn them directly into
individual resale value. Sports cards aren't cash, so the market has a lot of
friction. For the valuable cards, there's a whole rating system, based on the
condition of the card, and there's the 'work' of sleeving the cards and not
getting fingerprints on them. </p>
<p class="MsoNormal">If you want to get into cards as a money-making play, look
for an edge. See if you live an in area where there isn't a lot of 'flippers'
who buy up all the retail boxes. Look into leagues that you think could get
more attention later, like the WNBA (basketball), KHL (hockey), or NPB and KBO
(baseball). This could be your chance to make a long-term bet on the league
itself.</p>
<p class="MsoNormal">In a 67-card box (60 from the basic set plus 7 others) of
Topps Update 2020, which costs about $20 Canadian, the chances of getting
something special (e.g. a card with an alternate colour or photo, a signed
card, a limited run card, or a piece of memorabilia) is nearly 100%, with the less
special things being far more likely.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Even when you don't get anything worth a lot of money, you
still get 67 cards, and that's entertainment value. If sleeving and
categorizing cards isn't work to you, then you're not making an investment so
much as paying for entertainment with a small chance at a big payoff.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Which brings us to one last point of reference:
scratch-and-win lottery tickets.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">The scratch and win ticket "Crossword Extreme IV" costs
$20 Canadian as well, has a 40% chance of paying out anything, with the chances
of prizes dropping greatly at as prizes increase. (Details here: <a href="https://lotto.bclc.com/scratch-and-win/tickets/ticket-management/crossword-extreme-iv-310103.html">https://lotto.bclc.com/scratch-and-win/tickets/ticket-management/crossword-extreme-iv-310103.html</a>
<span style="mso-spacerun: yes;"> </span> )</p><p class="MsoNormal"></p><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp68sNJDCSFeft9n1Vl9PX2OqKKCoOQB_6VdT4ZCuSdi4nD85VXjLVwUpfUxXrVF-tNX-l59U3rvNJ5FKmiIMeP2jqTkt-LbdMWI7l_KMw2fDHk6_Ega-Y_vruoqycTzlgSoi_D5r19Z1w/s800/crossword-extreme-front-310103.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="800" data-original-width="760" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp68sNJDCSFeft9n1Vl9PX2OqKKCoOQB_6VdT4ZCuSdi4nD85VXjLVwUpfUxXrVF-tNX-l59U3rvNJ5FKmiIMeP2jqTkt-LbdMWI7l_KMw2fDHk6_Ega-Y_vruoqycTzlgSoi_D5r19Z1w/s320/crossword-extreme-front-310103.jpg" /></a></div><br /> <p></p><p class="MsoNormal"></p><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2UqekQkoSEKuZ1_dQIyDRpuh-Z7FY3nt0tkzRIFqCXmTV8_HmsU-L8d6aCCBlRSCB_I8BaXc_lvgGoIUUH-hC3yp2PeCmIxWkrdXYnDQC9G_AqxOlmpDDcfLGhS3b5e7OyefLUckBzD4q/s795/crossword-extreme-back-310103.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="795" data-original-width="764" height="341" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2UqekQkoSEKuZ1_dQIyDRpuh-Z7FY3nt0tkzRIFqCXmTV8_HmsU-L8d6aCCBlRSCB_I8BaXc_lvgGoIUUH-hC3yp2PeCmIxWkrdXYnDQC9G_AqxOlmpDDcfLGhS3b5e7OyefLUckBzD4q/w328-h341/crossword-extreme-back-310103.jpg" width="328" /></a></div><br /> <p></p><p class="MsoNormal"> <!--[if gte mso 9]><xml>
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</p><p class="MsoNormal">As an investment, it's terrible, but at least it's less work
to collect on the payout. As an entertainment product, personally I think of
scratch-and-wins as card packs with a much shorter entertainment time, and no
consolation prize of pictures of sports players if you don't win.</p>
<p class="MsoNormal">I chose this scratch-and-win game in particular because it
takes a lot of time to play compared to most scratch-and-wins, so you're either
getting the most entertainment for your money, or doing the most work for your
investment, depending on your perspective.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">--------------------------</p>
<p class="MsoNormal"><span style="font-size: 18pt; line-height: 107%;">Follow up
questions</span></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q1) If it costs $0.20 to get a random card, and $1 to buy
specific individual cards, at what point is it cheaper to start buying
individual cards than random cards? (Set of 50 cards, 1 set each, all equal
chance)</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q2) What distribution does the number of copies of each card
you have after a fixed number N of random cards drawn? Assume a set of size S
all with equal chance 1/S. (Answer: Binomial. p = 1/S, n=N)</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q3) As the number of random cards gets large, what does this
approximate? (Answer: Poisson, then eventually Normal)</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q4) Every additional set takes fewer additional cards to
complete, but there is a natural minimum to the expected number of cards to
complete an additional set. What is it and why? Assume every card is equally
likely. (Answer: 1/setsize, because that's the number of cards to get the last
card in the set, and we cannot start an additional set with fewer than 1 card
remaining)</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q5) How many cards would it take to complete a set with many
rares and one common card? Try this with 10 cards of .001 chance each, and 1
card of .99 chance. Is this what you would expect?</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q6) How many cards would it take complete a set with a
gradient of rarities? Try this with 10 cards of probability 1/55, 2/55, 3/55,
... , 10/55.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q7) For 300 equally likely cards, what is the pack size that
would require you to buy the fewest total cards to complete one set. Round to
the nearest pack size of 10.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q8) Panini sells packs of stickers with 2 stickers per pack
with no duplicates. In the 2018 World Cup set, there were 681 stickers in
total. Did selling stickers in packs of 2 reduce the total number of stickers
needed compared to buying them randomly 1 at a time?</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">Q9) The 2008 Magic: The Gathering set "Shards of
Alara" has 229 cards: 101 are 'common', 60 are 'uncommon', 53 are 'rare',
and 15 are 'mythic rare'.</p>
<p class="MsoNormal"> </p>
<p class="MsoNormal">In each pack of cards, there are 10 commons, 4 uncommons,
and 1 rare. In 1/8 of all packs, there is also a mythic rare. Assume there are
no duplicates in a pack. Assume that within each rarity, each card is equally
likely.</p>
<p class="MsoNormal">Q9a) How many packs of cards does it take to complete a set,
on average?</p>
<p class="MsoNormal">Q9b) How many packs of cards does it take to get 4 copies of
each card?</p>
<p class="MsoNormal">Q9c) How many packs of cards does it take to get 4 copies of
each card, excluding mythic rares?</p><p class="MsoNormal"> </p><p class="MsoNormal">Want more related content? This is part of <a href="https://www.stats-et-al.com/2020/07/statistics-gambling-and-games-of-chance.html">a coursepack on Statistics and Gambling</a> <br /></p>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Block Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 6"/>
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<![endif]-->Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-80084543301560539772021-01-21T17:48:00.000-08:002021-01-21T17:48:18.109-08:00Fantasy Sports Explained<p></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Information on fantasy sports ranges from the overly broad or impossible to apply like personal stories about how fantasy sports changed lives, to extremely narrow or short-lived like opinions on players recent performances. This article is intended to cover some of the middle ground to help you understand the basics of fantasy with a worked example of how you might choose players at the beginning of a league.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> <span></span></span></span></p><a name='more'></a><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"></span><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><i><span style="font-size: small;">This article may appear later in my coursepack "<a href="https://www.stats-et-al.com/2020/07/statistics-gambling-and-games-of-chance.html">Statistics, Gambling, and Games of Chance</a>". Here I mostly talk about fantasy leagues that last one or more seasons, as opposed to daily fantasy, which will be covered in a later article.</span></i></span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Before we start..</span></span></p><div style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; text-align: left;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span style="font-size: small;">If you want to try fantasy, just try it. Don't be intimidated. The best time to join a league is just before the start of a season, but public website have leagues starting all the time. You don't need to know the players, or even have any sports knowledge; the autopilot features on popular websites will make reasonable drafting decisions for you. Similarly, your team will play with or without your input, so you're not obligated to play through the season.</span></span></span><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></div><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Fantasy generally falls into two main categories called season long and daily fantasy.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Season long fantasy</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In season long fantasy, you take the role of a manager in a league of 5-15 teams. A season long fantasy experience is meant to mimic the experience of being an actual manager in the sport in question. Fantasy managers draft your players close to the beginning of the regular season of the real-life league.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">A player can only belong to a single manager at a time, they can be traded to other managers, or released to make room for other players. Sometimes there are restrictions on the number of trades to minimize the advantage of more active managers. Sometimes, there is a fantasy trade deadline that corresponds with the trade deadline in the real sport. Some leagues run across multiple years and managers keep the players they had drafted or traded for in previous years. Sometimes there are also limits on the salary that a fantasy team can have, where a fantasy salary is often based on the players' actual salaries.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In a formal draft, all the players in the league take turns selecting one player each for their team in rounds, usually in a 'snake format'. For example, consider a 10-team league where the manager that chooses the first player is Manager A, the second one is Manager B, ... , all the way to Manager J. The first round of ten players would go to managers A,B,C, ... , H,I,J. The next round of ten players would go to the managers in the opposite order: J,I,H, ... , C,B,A. So manager A would select the 1st player, 20th player, 21st player, 40th player, 41st player, and so on. Meanwhile, Manager C would select the 3rd, 18th, 23rd, 38th, and 43rd players.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">All of these restrictions and factors make player selection very complex, and it's best to go into a formal draft prepared with a pick list. A pick list is an ordered list of players that you want in a draft. When it's your turn to choose a player, you simply choose the highest-ranked player among those still available.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Adhering to the pick list isn't perfect because you might end up with too many of one position and not enough of another, or be stuck with a suboptimal choice for a given position. These considerations are relatively easy to account for during the draft just by tracking how many of each position you need. It's about opportunity cost, and there are tools online to do mock drafts to practice to get the most out of your pick list.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In season long fantasy, competition is head-to-head. Each week, managers decide which players they own will be active, meaning that their actions are the ones that count towards the fantasy team's score that week. A typical fantasy team has more players than they will actively use each week to account for player injuries; it's up to the manager to guess which of their players will perform best in any week. </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The score of a fantasy team is measured against another fantasy team in the league to determine which team plays that week. The fantasy teams with the best win-loss record often have a playoffs of a few weeks at the end of the regular season.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Season long - player scoring</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Player scoring in season long fantasy can take on either a points system or best-of-category system. In a points system, each event that a player does is worth a certain number of points. In hockey, a goal might be worth 10 points, an assist 8 points, a penalty could be worth 3 (or negative 3) points per minute. Scoring varies between sports and between fantasy leagues within a sport.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In a best-of-category system, stats are aggregated across active players. These stats could be simple events like number of goals or number of assists, or they could be more complex like plus/minus or save percentage. (or in, say, American football, pass completion percentage). Whichever fantasy team does better on the most categories in a week is the winner.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Daily fantasy</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In daily fantasy, a team is selected from all the players available for all events in a short period, usually a single day or tournament. There can be thousands of managers in a single daily fantasy competition. Participation in a daily fantasy competition is much simpler than in season-long fantasy competitions.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In daily fantasy, there is no formal draft process, there is no trading between players, and every player selected is actively used for that competition. Managers can enter at any time before the real sporting action starts, managers can select a player even if other managers have selected that player. Usually, the only player-selection restrictions are that managers can only select one copy of a player, that they need to fill a certain number of each role, and that they have keep their selections below a certain maximum salary.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Daily fantasy sports use a scoring-based system where every event is worth a certain number of points. Competition is a free-for-all instead of head-to-head, and prizes are often given to the top performers of the day. For example, a top prize might be given to highest scoring team, while the diminishing prizes might be given for lower places down until the top 10-20% are compensated.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">A simple strategy would be to simply pick the players with the highest expected number of points within the salary restrictions. However, since prizes tend to be heavily concentrated on the top performers, it often makes more sense to select players that are almost as good, but have been selected by a smaller proportion of players already entered. If you simply choose the best, most popular players (called a 'chalky' strategy in daily fantasy and a loosely related betting structure called a bracket), you are likely to maximize you expected score, but since your score will be strongly correlated with that of other fantasy managers, you will be unlikely to get a very high ranking.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 14pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Brackets and playoff drafts</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Brackets are a multiple-day version of fantasy that lasts as long as a playoff tournament. In brackets you have to predict which teams will move on at each around, and which ones will be eliminated. In playoff drafts, player selections are made similar to daily fantasy, but you stick with your player choices throughout the playoffs. Brackets and playoff drafts will be addressed in another article, but a good discussion of strategy can be found in "Optimal Drafting in Hockey Pools" at </span><a href="https://www.sfu.ca/~tswartz/papers/hockey.pdf" style="text-decoration: none;"><span style="background-color: transparent; color: #1155cc; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration-skip-ink: none; text-decoration: underline; vertical-align: baseline; white-space: pre-wrap;">https://www.sfu.ca/~tswartz/papers/hockey.pdf</span></a><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> by Amy E. Summers, Tim B. Swartz and Richard A. Lockhart.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 16pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Worked example: NHL Hockey on Yahoo! Sports.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><i>As a reminder, please don't be intimidated by the details here. This was just an attempt to approach fantasy scientifically and to repurpose data that had already been collected and processed for other projects. Your approach does not need to be anything close to this complicated, and the 'autopilot' features of popular websites can guide you through the drafting process just fine.</i><br /></span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Yahoo! Sports hosts free-to-play season long fantasy leagues for several sports, including NHL Ice Hockey. By default the league rules are set up as follows:</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">- There are 4-20 managers in a league, matches are head-to-head.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">- There is a formal draft before the regular season. Trades can happen up until real life trade deadline.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">- Scoring is based on the following 10 categories:</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">For skaters - Goals (G), Assists (A), Plus/Minus (PM), Powerplay Points (PPP), Shots on Goal (SOG), Hits (HIT)</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">For Goaltenders - Wins (W), Goals Against Average (GAA), Save Percentage (SV), Shutouts (SHO).</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">- Each team has 2 centers, 2 left wingers, 2 right wingers, 4 defenders, 2 goalies, 4 bench spots, and space to retain 2 injured players.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">In private leagues, changes to these rules can be made, such as including other categories like shorthanded goals, as well using a points-based system to score matchups instead of best of categories.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Figure 1 has the scores from a category-based league.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoeJ2ULdoFMTHnH_Tr3RqcOCRSvu-FYqcH9gjWwluT2uSF5snI71mlIHNfFBftQg7DE1mbDDjuZHpTCr883vpydrx93jEy6gLMT9nODAJDXvFDux7bFDR9brqsZ799Do1P1nika8RzLTta/s1005/Category+Matchup+Scoring.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="244" data-original-width="1005" height="111" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoeJ2ULdoFMTHnH_Tr3RqcOCRSvu-FYqcH9gjWwluT2uSF5snI71mlIHNfFBftQg7DE1mbDDjuZHpTCr883vpydrx93jEy6gLMT9nODAJDXvFDux7bFDR9brqsZ799Do1P1nika8RzLTta/w455-h111/Category+Matchup+Scoring.PNG" width="455" /></a></span></div><span style="font-family: arial;"><br /><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span id="docs-internal-guid-df1db051-7fff-b857-dc31-9626680f1b44" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Figure 2 has the scores from a points-based league, and Figure 3 has an example points breakdown for a player.</span></span></span><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span id="docs-internal-guid-df1db051-7fff-b857-dc31-9626680f1b44" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPnlUs0i16fcjAtPCJmtifSNXKQb8U9jqUAvEUPOpT8uooRfh_P1HqhLOf_B8wH9d_iUizDM9A0ZUOYC3CwFuhyphenhyphenYOpjkNpdRi_Acb55zN1T0Zssjpcio4SKvLXJrhVbSkzGNa-ST4vXIKt/s1068/Fanpoints+Matchup+Scoring.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="415" data-original-width="1068" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPnlUs0i16fcjAtPCJmtifSNXKQb8U9jqUAvEUPOpT8uooRfh_P1HqhLOf_B8wH9d_iUizDM9A0ZUOYC3CwFuhyphenhyphenYOpjkNpdRi_Acb55zN1T0Zssjpcio4SKvLXJrhVbSkzGNa-ST4vXIKt/w462-h180/Fanpoints+Matchup+Scoring.PNG" width="462" /></a></span></div><span style="font-family: arial;"><br /></span><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span id="docs-internal-guid-df1db051-7fff-b857-dc31-9626680f1b44" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNlxWKpVVayfvulT5OFepnRIjc4oxuULuyNSC-ozv-d9q6uh_ZGj6yKsHbpJu9ZtVqclddsNzFTIMtbq_XKyshmpPCBiPwKMRsPR0ufn-GSzp9jQ9Yo9X5blDZxJsCukFPf7hqSbMPkcn7/s323/Fanpoints+Breakdown.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="279" data-original-width="323" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNlxWKpVVayfvulT5OFepnRIjc4oxuULuyNSC-ozv-d9q6uh_ZGj6yKsHbpJu9ZtVqclddsNzFTIMtbq_XKyshmpPCBiPwKMRsPR0ufn-GSzp9jQ9Yo9X5blDZxJsCukFPf7hqSbMPkcn7/s320/Fanpoints+Breakdown.PNG" width="320" /></a></span></div><span style="font-family: arial;"><br /></span><p dir="ltr" id="docs-internal-guid-4cfe9307-7fff-fa55-bb70-e954fb3acad0" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Each week, you pick players from your team, and their aggregate in each of those categories determines who wins. So a manager with better G, A , PM, SOG, HIT, and W in a week would get 6 points, beating their opponent who can get at most 4 points from PPP, GAA, SV, and SHO.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">To prepare, I've used Sportlogiq's data to make some picklists. I assigned each player a value based on these ten categories. Skaters will get 0 for the goalie-based categories. Some are simple (Goal value is based on the number of goals a player gets), others are trickier (Plus-Minus is based on defensive measures like dumpouts, shots and passes blocked, and shots interfered with)</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">I've also assigned players a "soft" value based on playmaking events that don't fit any specific category. These are events like faceoffs, loose puck recoveries, controlled zone entries, and (negative) penalties received. These are probably useful, but it's hard to stick them to a particular category.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">For goalies, "wins" and "shutouts" are partially based on the general ability of the team that the goalies play for. This is because they're partially a reflection of the team's ability to produce goals and avoid shots against.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">To prevent the effect of players only appearing for a few good games, and to make sure my players would be available to play most of the time, I scaled all these values based on the number of games in the last three regular seasons that each player had appeared in.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The formula that I used to evaluate skating players was:</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Player Value = sqrt(Regulation games in last 3 years) * [beta_G * Goal value + beta_A *Assist value + beta_PP *Powerplay value + beta_PM *Plus-Minus value + beta_SOG * Shots Value + beta_HIT *Hits Value + beta_S * Soft value]</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">where</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Goal value = Goals / game</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Assist value = 1/3 * (puck protections + dumpins + passes) / game</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Plus-minus value = 1/4 * (dumpouts + shots blocked + passes blocked + shots pressured) / game</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Powerplay value = goal value + assist value – 5v5 goals</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Shots value = 1/3 * (2*shots on goal + shot attempts) / game</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Hits value = Checks /game</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Soft value = 1/2 * (loose puck recoveries + blue line carries + pass receptions + faceoff wins – faceoff losses – penalies) / game</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">and where</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Beta_G, Beta_A, beta_PP, ... are normalizing factors based on league averages. For example, there are 10-20 hits for every goal in a game, so beta_G is 10-20 times as large as beta_HIT, and a goal counts for 10-20 times as much in player valuation.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The formula that I used to evaluate goalies was:</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Player Value = sqrt(Regulation games on roster, last 3 years) * [beta_W * Wins value + beta_GAA *Goals against value + 2*beta_SV *Save Percentage value + beta_SHO * Shutout value]</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">where</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Wins value = Elo-rating of goalie's team</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Goals against value = 1 / goalie's goals against average</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Save percentage value = goalie's saves / goal</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Shutout rating = goalie's shutout percentage, last 40 starts</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">and where</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Beta_W, Beta_GAA, beta_SV, and beta_SHO are normalizing factors based on league averages.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Note that save percentage counts for twice as much as every other value because it's the only one that is reflection of only the goalie and not the team they play for.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">After I applied these formulas for each player, I applied weights to each factor (on top of the normalizing factors) to get got an aggregate value for each player. I don't know in advance what the weightings should be, so I messed with them ad-hoc until the 'tiers' of the pick list resembled the 'average draft position' (ADP) across several popular fantasy websites, including Yahoo! Sports. I found these at</span><a href="https://www.fantasypros.com/nhl/adp/overall.php" style="text-decoration: none;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span><span style="background-color: transparent; color: #1155cc; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration-skip-ink: none; text-decoration: underline; vertical-align: baseline; white-space: pre-wrap;">https://www.fantasypros.com/nhl/adp/overall.php</span></a><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> . </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">The aim here isn't to replicate last year's draft positions, but to approximate the relative importance of forwards, defenders, and goalkeepers.</span></span></p><span style="font-family: arial;"><br /></span><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Going from tiers of 50 players each, the crowd picks positions with the following breakdown.</span></span></p><div align="left" dir="ltr" style="margin-left: 0pt;"><table style="border-collapse: collapse; border: none;"><colgroup><col width="149"></col><col width="154"></col><col width="153"></col><col width="151"></col></colgroup><tbody><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Tier</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Defenders</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Forwards</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Goalies</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1st – 50th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">8</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">34</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">8</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">51st – 100th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">13</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">28</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">9</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">101st – 150th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">18</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">27</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">5</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">151st – 200th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">19</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">24</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">7</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">201st – 250th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">12</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">34</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4</span></span></p></td></tr></tbody></table></div><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">My weighted sums based on raw event counts produced this:</span></span></p><div align="left" dir="ltr" style="margin-left: 0pt;"><table style="border-collapse: collapse; border: none;"><colgroup><col width="149"></col><col width="154"></col><col width="153"></col><col width="151"></col></colgroup><tbody><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Tier</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Defenders</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Forwards</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Goalies</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1st – 50th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">40</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">51st – 100th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">10</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">37</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">3</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">101st – 150th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">14</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">30</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">151st – 200th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">17</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">25</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">8</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">201st – 250th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">15</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">29</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6</span></span></p></td></tr></tbody></table></div><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">My weighted sums based on 'proportion of team's output' produced this:</span></span></p><div align="left" dir="ltr" style="margin-left: 0pt;"><table style="border-collapse: collapse; border: none;"><colgroup><col width="149"></col><col width="154"></col><col width="153"></col><col width="151"></col></colgroup><tbody><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Tier</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Defenders</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Forwards</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Goalies</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1st – 50th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">40</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">51st – 100th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">9</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">37</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">101st – 150th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">12</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">21</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">7</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">151st – 200th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">17</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">23</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">10</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">201st – 250th</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">17</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">29</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4</span></span></p></td></tr></tbody></table></div><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Also, the top ten picks of each method and the crowd look like this</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><div align="left" dir="ltr" style="margin-left: 0pt;"><table style="border-collapse: collapse; border: none;"><colgroup><col width="82"></col><col width="165"></col><col width="180"></col><col width="180"></col></colgroup><tbody><tr style="height: 39.25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Rank</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Crowd</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Raw Output Method</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">% of Team Output Method</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Nikita Kucherov RW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Alex Ovechkin LW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Leon Draisaitl C</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">2</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Connor McDavid C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Connor McDavid C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Connor McDavid C</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">3</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Alex Ovechkin LW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Nikita Kucherov RW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Alex Ovechkin LW</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Nathan MacKinnon C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Auston Matthews C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Jack Eichel C</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">5</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Patrick Kane RW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Leon Draisaitl C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">David Pastrnak RW</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Brad Marchand LW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Nathan MacKinnon C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Auston Matthews C</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">7</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">David Pastrnak RW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">David Pastrnak RW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Mika Zibanejad C</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">8</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Sidney Crosby C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Patrick Kane RW</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Anze Kopitar C</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">9</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Brent Burns D</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">John Tavares C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Patrick Kane RW</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">10</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Leon Draisaitl C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Jack Eichel C</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Nathan MacKinnon C</span></span></p></td></tr></tbody></table></div><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Finally, the distribution of values by rank is shown in Figure 4. Here, a few players at the top are much more valuable than the rest, implying that the opportunity cost of missing one of these elite players in the first few rounds is much more important than position-balancing considerations, which can be done in later rounds when the drop off in output is not so steep.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"><span id="docs-internal-guid-df1db051-7fff-b857-dc31-9626680f1b44" style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></span></p><div class="separator" style="clear: both; text-align: center;"><span style="font-family: arial;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK7ntASdgvkuS-SqQFMUE_Otfq24WqOGryvLYNgwmsd_yZyElX_ahb53_lzEVg6ZATncgklhf7Zf7qt4vM8QyzEX26sGwrV3gE0Kdr8OBOLdlD-63PdEQO0cjEZBVi4bmBQWCEDUhhUl31/s624/Value+by+Rank.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="255" data-original-width="624" height="187" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK7ntASdgvkuS-SqQFMUE_Otfq24WqOGryvLYNgwmsd_yZyElX_ahb53_lzEVg6ZATncgklhf7Zf7qt4vM8QyzEX26sGwrV3gE0Kdr8OBOLdlD-63PdEQO0cjEZBVi4bmBQWCEDUhhUl31/w457-h187/Value+by+Rank.png" width="457" /></a></span></div><span style="font-family: arial;"><br /></span><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" id="docs-internal-guid-c97ae92b-7fff-d60f-5bba-3d7d1fa96882" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">We’ll see as the league progresses how well this all works, it’s just mostly a toy example of what’s possible to show how it all works.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> I opted for the raw output method for three different leagues.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">"Pronogistotron Alpha" plays in a public league of 12 managers. I drafted 1st (then 24th, 25th, 48th,…).</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">"Pronogistotron Beta" plays in a private league of 8 managers among friends. I drafted 1st (then 16th, 17th, 32nd,…).</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">"Pronogistotron Gamma" plays in a private league of 14 managers among Sportlogiq colleagues. I drafted 12th (then 16th, 40th, 44th,…).</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">As "Alpha", I took the highest-valued player that was available until all position requirements were filled, then I chose the highest-valued player left. Also, I accidentally autodrafted in my first round as I was trying to learn the system.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">As "Beta", I used the system, but also avoided injured players, and lowered the priority for defensive players to try and adapt to the league's scoring system.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">As "Gamma", I used the updated system, but also tried to avoid backup goalies to make sure I had goalies who actually played each week. I also sought players on strong teams to improve my plus-minus and assist results.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Two goalies, Allen and Grubauer, are in all three leagues (bold and italic), these represent the picks that my system rates substantially higher than other managers' systems.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><div align="left" dir="ltr" style="margin-left: 0pt;"><table style="border-collapse: collapse; border: none;"><colgroup><col width="75"></col><col width="179"></col><col width="166"></col><col width="188"></col></colgroup><tbody><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Picks</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Alpha</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Beta</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Gamma</span></span></p></td></tr><tr style="height: 39.25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">1</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">McDavid (EDM – C)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Ovechkin (WAS – LW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">P. Kane (CHI – RW)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">2</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Kucherov (TB – RW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">P. Kane (CHI – RW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Tavares (TOR – C)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">3</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Pastrnak (BOS – RW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Tavares (TOR – C)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Kopitar (LA – C)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">4</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Barkov (FLA – C)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Malkin (PIT – C)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Markstrom (CGY – G)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">5</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: italic; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Allen (MTL – G)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: italic; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Allen (MTL – G)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: italic; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Grubauer (COL – G)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">6</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: italic; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Grubauer (COL – G)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: italic; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Grubauer (COL – G)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">E. Kane (SJ – LW)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">7</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Letang (PIT – D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Giroux (PHI – LW, C)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Letang (PIT – D)</span></span></p></td></tr><tr style="height: 39.25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">8</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Benn (DAL – C, LW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Pastrnak (BOS – RW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">DeBrincat (CHI – LW,RW)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">9</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Giordano (CGY – D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Kopitar (LA – C)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Karlsson (VGK – C)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">10</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Ehlers (WPG – LW, RW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Werenski (CBJ – D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Benn (DAL – C, LW)</span></span></p></td></tr><tr style="height: 39.25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">11</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Doughty (LA – D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Ekman-Larsson (ARI – D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Spurgeon (MIN – D)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">12</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Subban (NJ – D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Doughty (LA – D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Rakell (ANA – LW, RW)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">13</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Rakell (ANA – LW, RW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Barrie (EDM – D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: italic; font-variant: normal; font-weight: 700; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Allen (MTL – G)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">14</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Zucker (PIT – LW, RW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Fleury (VGK – G)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Ekholm (NSH – D)</span></span></p></td></tr><tr style="height: 39.25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">15</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Ekholm (NSH - D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Dadonov (OTT – LW, RW)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Gustafsson (PHI- D)</span></span></p></td></tr><tr style="height: 25pt;"><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">16</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Jones (SJ – G)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Gustafsson (PHI- D)</span></span></p></td><td style="border-bottom: solid #000000 1pt; border-left: solid #000000 1pt; border-right: solid #000000 1pt; border-top: solid #000000 1pt; overflow-wrap: break-word; overflow: hidden; padding: 5pt 5pt 5pt 5pt; vertical-align: top;"><p dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Faulk (STL – D)</span></span></p></td></tr></tbody></table></div><span style="font-family: arial;"><br /></span><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">It's early in the season, so results are minimal. However, there are a few other improvements to be made before taking this seriously, or to consider before tackling daily fantasy sports.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; padding: 0pt 0pt 0pt 18pt; text-indent: -18pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">· Customizing player valuation for each league. Not just to the rules (e.g. points vs categories).</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; padding: 0pt 0pt 0pt 18pt; text-indent: -18pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">· Making penalties their own value apart from "soft", so that their value could be either negative, positive, or neutral, depending on the rules of the league.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; padding: 0pt 0pt 0pt 18pt; text-indent: -18pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">· Including more events directly, like assists, plus-minus and shorthanded goals, there are some other improvements to be made.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; padding: 0pt 0pt 0pt 18pt; text-indent: -18pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">· Evaluations for plus-minus should account for the team that a player is on.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; padding: 0pt 0pt 0pt 18pt; text-indent: -18pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">· Discounts applied based on a player's injury status at the beginning of the year. A player that is going to be out of commission for a few months looks exactly the same as any other player here.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; padding: 0pt 0pt 0pt 18pt; text-indent: -18pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">· The number of games a player starts as a goalie isn't considered.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; padding: 0pt 0pt 0pt 18pt; text-indent: -18pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">· Considerations for players that fit into more than one role, like a LW/RW.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt; padding: 0pt 0pt 0pt 18pt; text-indent: -18pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">· Opportunity cost based on position isn't considered yet. In other words, how much are you losing by leaving a player unpicked until the next round? It's a lot if there's nobody as good as them left, and not much if there are many players like them remaining. This is probably much more complicated than other improvements on this list.</span></span></p><p><span style="font-family: arial;"><br /></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Bibliography:</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Episode 01.26 of the Too Many Men podcast and Alexandra Mandrycky of the Seattle Kraken for their insights into drafting.</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">Fantasy Life by Matthew Berry</span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;"> </span></span></p><p dir="ltr" style="line-height: 1.38; margin-bottom: 12pt; margin-top: 12pt;"><span style="font-family: arial;"><span style="background-color: transparent; color: black; font-size: 11pt; font-style: normal; font-variant: normal; font-weight: 400; text-decoration: none; vertical-align: baseline; white-space: pre-wrap;">I used Sportlogiq data here, but given that this isn't a very deep or high-quality analysis, you could recreate something just as good or better with the data that MoneyPuck offers for free at <a href="http://www.moneypuck.com/data.htm">http://www.moneypuck.com/data.htm</a></span></span></p>Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-27462763311341913712020-12-20T20:26:00.001-08:002020-12-21T00:13:34.541-08:00Borel Dice Edition - Brute Forcing Experiments<p> <!--[if gte mso 9]><xml>
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</p><p class="MsoNormal" style="margin-bottom: 0cm;"><a href="https://www.playborel.com/">Borel and Borel: Dice Edition</a> are
educational games about probability. I picked up a copy of each because I
thought they would be useful in introducing some ideas of <a href="https://www.stats-et-al.com/2020/07/statistics-gambling-and-games-of-chance.html">probability and gambling</a> without the cultural baggage of better known games.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">I'm biased because it's my
field, but Borel has a lot more play value than most games of its kind. The
dice edition, which is much easier to find, and easier to get into and play,
has a set of 7 dice (four 6-sided, and one each of a 10-sided, 20-sided and
30-sided die), and a deck of 100 "experiments", like Experiment 001:</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> <span></span></p><a name='more'></a><p></p>
<div class="MsoNormal" style="margin-bottom: 0cm; text-align: left;"><span face=""Candara",sans-serif" style="font-size: 14pt; line-height: 107%;"> <i>This post does not contain a paid affiliate link, nor was I paid to write it. However, I did receive both Borel and Borel: Dice Edition from the games' creators for only the price of shipping.</i></span><b><span face=""Candara",sans-serif" style="font-size: 14pt; line-height: 107%;"><i><br /></i></span></b></div><p class="MsoNormal" style="margin-bottom: 0cm;"><b><span face=""Candara",sans-serif" style="font-size: 14pt; line-height: 107%;"> </span></b></p><div class="separator" style="clear: both; text-align: center;"><b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDOxvl7qdcYXQgd6M8yE1DrDLpObwGSGCI36SuyMto-dPaZqALOUj8yUD45WGBw0wkq03eLG6ZkbS4OuUmD04BpP8q5mvtM9B9vs8-YfwVY0Cz_8yRaY8gBKUywJ3Q3Ze-felunBdbZuhz/s859/Borel+Box.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="859" data-original-width="617" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDOxvl7qdcYXQgd6M8yE1DrDLpObwGSGCI36SuyMto-dPaZqALOUj8yUD45WGBw0wkq03eLG6ZkbS4OuUmD04BpP8q5mvtM9B9vs8-YfwVY0Cz_8yRaY8gBKUywJ3Q3Ze-felunBdbZuhz/s320/Borel+Box.PNG" /></a></b></div><b><br /></b><p></p><p class="MsoNormal" style="margin-bottom: 0cm;"><b><span face=""Candara",sans-serif" style="font-size: 14pt; line-height: 107%;">Experiment 001: Roll d6, d6,
d6, d6, d10, and d30. Will you roll 5 or more unique numbers?</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Each player bets 1 point on
whether the stated outcome will happen or not, so the winner of a game (of
10-20 experiments) is whoever guesses the outcome the most often. </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The experiments are designed so
that outcomes are close to 50-50. but not quite. If you have a sense of
probability, you can work out the more likely outcome by intuition sometimes.
However, the experiments are not so obvious that you can scratch out the
probability by hand in a minute.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">For example, in Experiment 001,
finding the distribution of the number of uniques for identical dice isn't
hard.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">(e.g. Pr(3 or more uniques) = 6/6
* 5/6 * 4/6 * 6/6)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">But once you introduce two
additional dice of different numbers of sides, the computation gets too
complicated for a casual tabletop game. That's the point though, if you enjoy
math puzzles, or reading chance magazine or anything by Martin Gardner, then
the experiments are deep enough to have nerd gravity. If you just want to play
a short casual game with dice and more variety than Yahtzee, then you still
have a shot at winning with thinking too hard.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">You're not supposed to prepare, it
sort of ruins the game. You can't have "Ruin" without "R".</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">First, let's make a function to
roll dice. Not a perfectly efficient one, but a readable, informative one.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">random.dice
= function(dice_used = c(6,6,6,6,10,20,30), Nrolls=100000, seed=NA)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">if(!is.na(seed)){set.seed(seed)}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Ndice
= length(dice_used)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat
= matrix(NA, nrow=Nrolls, ncol=Ndice)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">for(k
in 1:Ndice)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat[,k]
= sample(1:dice_used[k], size=Nrolls, replace=T)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">return(rollmat)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Test: Outputs a 10 x 7 matrix with 10 random rolls of 4d6, d10, d20, d30</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
random.dice(Nrolls=10)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">and because we can, let's also
make a function to generate one of every possibility</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">exhaustive.dice
= function(dice_used= c(6,6,6,6,10,20,30))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Ndice
= length(dice_used)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nrolls
= prod(dice_used)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;">rollmat = matrix(NA, nrow=Nrolls,
ncol=Ndice)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span lang="PT-BR" style="font-family: "Courier New"; mso-ansi-language: PT-BR;"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">for(k
in 1:Ndice)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Ntimes
= prod(dice_used[1:k]) / dice_used[k] </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Neach
=<span style="mso-spacerun: yes;"> </span>prod(dice_used[k:Ndice]) /
dice_used[k] </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat[,k]
= rep(1:dice_used[k], times=Ntimes, each=Neach)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">return(rollmat)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Test: Outputs a 24 x 3 matrix with one of each possible combination of d2, d3,
d4</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
exhaustive.dice(dice_used = c(2,3,4))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The exhaustive.dice() function can
be used in cases that that are simple enough to get every possibility, but the
memory required scales exponentially with the number of dice rolled, so it
won't be availble if we have to do a lot of rolls for one experiment. [Foot 1]
We can use random.dice() to get something close to the true probability, and
can sometimes use exhaustive.dice() to check our work. [Foot 2] </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Finally, the seed argument in
random.dice() is optional. If you don't enter anything, you'll get a new random
set of numbers each time. If you do choose a seed, you can reproduce the same
numbers each time. So you can check your work easier, these notes will use an
easy to remember seed: my luggage password.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">A lot of these experiments can be
solved the same way: Generate some rolls, and apply a function row-by-row to
check of each row would be a 'success' or not. What changes each time are the
generator inputs and the function being applied.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">In this first experiment, we want
to know the number of unique numbers from each roll of the experiment.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">count.uniques
= function(roll)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">{</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"><span style="mso-tab-count: 1;"> </span>output = length(unique(roll))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">}</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nrolls
= 100000</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat
= random.dice(dice_used = c(6,6,6,6,10,30), Nrolls=Nrolls, seed=12345)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nuniques
= apply(rollmat, 1, count.uniques)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">##
The distribution</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">table(Nuniques)<span style="mso-spacerun: yes;"> </span>/ nrow(rollmat) </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#<span style="mso-spacerun: yes;"> </span>1<span style="mso-spacerun: yes;">
</span>2<span style="mso-spacerun: yes;"> </span>3<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;">
</span>5<span style="mso-spacerun: yes;"> </span>6 </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
0.00001 0.00285 0.06233 0.33397 0.46277 0.13807 </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">##
Pr(more than 4 uniques)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">length(which(Nuniques
> 4))<span style="mso-spacerun: yes;"> </span>/ nrow(rollmat) </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
<span style="mso-spacerun: yes;"> </span>0.60084</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Here's the histogram of the
simulated number of uniques</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">hist(Nuniques,
col="#4499BB", n=7, main="Experiment 001 - 100k
simulations")</span></p>
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<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Mention"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Smart Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hashtag"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Unresolved Mention"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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</span><p></p><p class="MsoNormal" style="margin-bottom: 0cm;">For experiment 1, there's
about a 60.0% chance of success. How close are we to the true probability with
100,000 rolls of the 6 dice of the experiment?</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat
= exhaustive.dice(dice_used = c(6,6,6,6,10,30)) # 388,000 possibilities</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nuniques
= apply(rollmat, 1, count.uniques)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">table(Nuniques)
/ nrow(rollmat) </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">length(which(Nuniques
> 4)) / nrow(rollmat) </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
<span style="mso-spacerun: yes;"> </span>1 <span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;">
</span>3<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;"> </span>5<span style="mso-spacerun: yes;">
</span>6 </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
0.000015 0.002886 0.063765 0.333333 0.461111 0.138889 </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
0.6</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We were within 0.1% of the true
value. Not bad! Let's tackle experiments 002 through 007 with these tools.</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p><p class="MsoNormal" style="margin-bottom: 0cm;"><!--[if gte mso 9]><xml>
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</p><p class="MsoNormal" style="margin-bottom: 0cm;"><b><span face=""Candara",sans-serif" style="font-size: 14pt; line-height: 107%;">Experiment 002: Roll d6, d6,
d6, d6, d10, d20, and d30. Will the sum be of the results be 45 or more?</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Intuitively, what should we
expect?</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The expected score of an N-sided
die roll is (1 + 2 + ... + N) / N = (N + 1) / 2</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We have 4 6-sided dice (Expected
score = 3.5 * 4 = 14)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We have 1 10-sided die (Expected
score = 5.5)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We have 1 20-sided die (Expected
score = 10.5)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We have 1 30-sided die (Expected
score = 15.5)</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Total E[score] = 45.5</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">So the average score is more than
45, and each die roll has a symmetric distribution, so the sum should be
symmetric too [Foot 3], so logically more than 50% of the probability is at 45
or more.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Is our intuition right?</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">Nrolls
= 100000</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat
= random.dice(dice_used = c(6,6,6,6,10,20,30), Nrolls=Nrolls, seed=12345)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sums
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<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
Plot out the distribution of the sum</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">hist(sums,
col="#4499BB", n=50, main="Experiment 002 - 100k
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<p class="MsoNormal" style="margin-bottom: 0cm;"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjExDU6uvvW4vnXD17wZ73Eg2ikn5h18W312mmxdRL8GpkuRYRWpjGbbIMDlGZ7WZEpKJpZu36TlQAP4SYLv0ObXKtCmvOMi6k02M-WZ-U2IQpinW6UX7KFtrlCDCi2DAZ8mymcBUhws9vZ/s847/Borel+Experiment+002+stats-et-al.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="747" data-original-width="847" height="353" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjExDU6uvvW4vnXD17wZ73Eg2ikn5h18W312mmxdRL8GpkuRYRWpjGbbIMDlGZ7WZEpKJpZu36TlQAP4SYLv0ObXKtCmvOMi6k02M-WZ-U2IQpinW6UX7KFtrlCDCi2DAZ8mymcBUhws9vZ/w400-h353/Borel+Experiment+002+stats-et-al.png" width="400" /></a></div><p></p><p class="MsoNormal" style="margin-bottom: 0cm;"><!--[if gte mso 9]><xml>
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<![endif]--></p><p class="MsoNormal" style="margin-bottom: 0cm;">
</p><p class="MsoNormal" style="margin-bottom: 0cm;">If that histogram shape looks
familiar, that's because it's a normal curve, which we would expect from a sum
of independent values. these aren't <i>identically distributed</i> independent values,
so it's not quite a normal, but at a casual glance, it's pretty close.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">length(which(sums
>= 45)) / nrow(rollmat)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
0.52988</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">So the random rolls showed a
success 52.99% of the time. The exact answer is going to get a minute or two to
compute, but it's doable.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat
= exhaustive.dice(dice_used = c(6,6,6,6,10,20,30))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">sums
= apply(rollmat, 1, sum)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">length(which(sums
>= 45)) / nrow(rollmat)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
0.5323611</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><b><span face=""Candara",sans-serif" style="font-size: 14pt; line-height: 107%;">Experiment 003: Roll d30, d30,
and d30. Will the highest result be greater than the lowest result by 16 or
more? </span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Before we jump into the code, what
are we expecting here? Consider the standard uniform distribution for a bit.
That's the continuous distribution where every value from 0 to 1 is equally
likely.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">The expected value from a uniform
is 1/2. [Foot 4]</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">If we take two values from a
uniform, the expected value of the higher value is 2/3, and the lower value is
1/3. [Foot 5]</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">If we take three values from a
uniform, the expected value of the highest one is 3/4, the expected value of
the lowest one is 1/4, and the expected value of the middle one is 1/2.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Applying this principle the values
of a single die roll, which follow a discrete uniform, we might assume that
after rolling three 30-sided dice that lowest value would be near 7.5, and the
highest value would be near 22.5, making a difference near 15. </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">There's a few issues though.
First, the expected values would be a little higher because the range is not
0-30, it's 1-30. Next, the distribution of a roll isn't continuous, it's
discrete, so all if the probability is on a finite number of values, including
the two extreme values of 1 and 30.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">We could estimate the success
probability to be below 50%, but it's not immediately clear which side of 50%
the probability is.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">Let's find out. First by
simulation, then exhaustively.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat
= random.dice(dice_used = c(30,30,30), Nrolls=Nrolls, seed=12345)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">maxs
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<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">mins
= apply(rollmat, 1, min)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">length(which((maxs-mins)
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<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
0.47381</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#Here's
the histogram</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">hist((maxs
- mins), col="#4499BB", n=30, main="Experiment 003 - 100k
simulations")</span></p>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Block Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 6"/>
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<p></p><p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">rollmat
= exhaustive.dice(dice_used = c(30,30,30))</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">maxs
= apply(rollmat, 1, max)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">mins
= apply(rollmat, 1, min)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">length(which((maxs-mins)
>= 16)) / nrow(rollmat)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><span style="font-family: "Courier New";">#
0.4744444</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">47.44% is farther below 50% than I
would have though.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">But tis the season to take a break
from experiments.</p><p class="MsoNormal" style="margin-bottom: 0cm;"> </p><p class="MsoNormal" style="margin-bottom: 0cm;"></p><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrn4Jfj0Lnpm9Z5CMzW9kAWqbvJmabiLJQ9ECemRaGPXsD6OPqXhPsIocCxJODsEDYIXjmrRiVWt_LiQ2v0nI7Z0_gl8wH_VLzt2aMJ6TT2CLksrrcPz4Z1LAj6ZFUcyJMhISEMk173gjo/s1600/chica+papa+noel+2020.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrn4Jfj0Lnpm9Z5CMzW9kAWqbvJmabiLJQ9ECemRaGPXsD6OPqXhPsIocCxJODsEDYIXjmrRiVWt_LiQ2v0nI7Z0_gl8wH_VLzt2aMJ6TT2CLksrrcPz4Z1LAj6ZFUcyJMhISEMk173gjo/w400-h300/chica+papa+noel+2020.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Don't eat that Santa all in one go.<br /></td></tr></tbody></table><br /> <!--[if gte mso 9]><xml>
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<p></p><p class="MsoNormal" style="margin-bottom: 0cm;">Borel: Dice Edition has 100
cards. The original Borel has more randomization elements like coins and
marbles in bags, as well as more complex betting elements like redos and
different betting amounts, but The Dice Edition is much more approachable. The
experiment cards can easily be modified or converted into assignment questions
for a statistics course, and you'll find more experiment breakdowns like this
in <a href="https://www.stats-et-al.com/2020/07/statistics-gambling-and-games-of-chance.html">Statistics and Gambling</a> when it's done.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">[Foot 1] Just using the seven base
dice once each takes a fair bit of memory, (7.78 million rows x 7 columns x ~4
bytes) = ~200 MB. </p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">[Foot 2] The binomial
approximation is also an option to see how far off the market random.dice() is
likely to be, and we can always repeatedly use random.dice() to get higher
accuracy.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">[Foot 3] This (sum of symmetric =
symmetric) property applies to sums and means, but it usually doesn't work for
other functions like products.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"> </p>
<p class="MsoNormal" style="margin-bottom: 0cm;">[Foot 4] Actually the distribution
from a single die roll is the discrete version of the uniform. If it helps, you
can think of uniform coming from a million-sided die. Using what we learned in
Experiment 002, the expected roll from a d1,000,000 would be 500000.5, or
almost exactly half a million. So seeing an expected value of 1/2 for a uniform
shouldn't be too surprising.</p>
<p class="MsoNormal" style="margin-bottom: 0cm;"><br /></p>
<p class="MsoNormal" style="margin-bottom: 0cm;">[Foot 5] This isn't saying that
each of the two values we took from a uniform is different somehow, just after
we sorted them, one value is going to be larger than the other. Just knowing
that it's the larger of the two gives us enough information to move the
expected value up from 1/2 to 2/3.</p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-11378018520108373212020-11-11T12:44:00.002-08:002020-11-11T13:37:39.548-08:00T1B: Goodhart's Law and Baserunning<span style="font-family: helvetica;">When I was about 11 years old, I was good at running, good
enough to represent my elementary school as the anchor in a relay at a district
track meet. This prompted all the grown-ups in my life to coach me on running. They
told me all sorts of tricks about keeping my hands flat and remembering to
breathe and to keep in a straight line and to keep from dragging my feet and to
start early to get the baton.
</span><p class="MsoNormal"><span style="font-family: helvetica;">The race came and I did all those things – hands, breathing,
straight line, no dragging, start early. I forgot, however, to run. I blew a
huge lead by running in perfect form, but in slow motion.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">What's the lesson here?</span></p><span style="font-family: helvetica;">
<span><a name='more'></a></span></span><p class="MsoNormal"><span style="font-family: helvetica;"><br /></span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">I mean, besides that I have low kinesthetic intelligence (I'm
slow to learn how to use my body). </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">What's a more general lesson? </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">Don't let minutia get in the way of winning.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"> </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">Often when there's resistance from coaches and players to something
a quant suggests, this is why. It's the sporting equivalent of Goodhart's Law,
or <i>"When a measure becomes a target, it ceases to be a good measure."</i>
<span style="mso-spacerun: yes;"> </span>[1] </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">So are stats worthless for coaching purposes in sports? Not
at all, it just means you need to consider the possibility of second-order
effects before you rely too heavily on them. Shot attempts is a great indicator
of hockey success until your players start shooting from center ice. On-base
percentage was at the core of the Moneyball strategy. Aiming for OBP gets more walks and uses more pitches, thereby using up a team's best pitchers faster. That strategy
works great right up until opposing pitchers learned to throw strikes more
often.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">What makes a good stat to measure and aim for? If you're a coach.
One with minimal opportunity to reach through some objectively bad strategy,
and one with minimal opportunity for counter play. </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">A good stat for player evaluation is also actionable in that
a player can do something to improve it; it should be reliably measurable and
attributable to the individual player. Finally, and most importantly, it has to
be correlated with success.</span></p><p class="MsoNormal" style="text-align: center;"></p><div class="separator" style="clear: both;"><span style="font-family: helvetica;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_-Q54bTKV12RSYEgQoEyHH42VKPFvHpHtWGur8b1EdQVsfECAqUtEaSY2UE6EfXKi2rfvhgsva6GbvZtnzo1GU9Q9Sr_3xQtMGCi-AGlQFXrby20OIRTHQ3U9Fst2Ei38O2SGeKNjD754/s640/Naruto-Run.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="300" data-original-width="640" height="174" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_-Q54bTKV12RSYEgQoEyHH42VKPFvHpHtWGur8b1EdQVsfECAqUtEaSY2UE6EfXKi2rfvhgsva6GbvZtnzo1GU9Q9Sr_3xQtMGCi-AGlQFXrby20OIRTHQ3U9Fst2Ei38O2SGeKNjD754/w372-h174/Naruto-Run.png" width="372" /></a></span></div><span style="font-family: helvetica;"><a href="https://hokagestore.com/scientists-confirm-that-running-like-naruto-would-make-you-200-faster/">200 percent faster? Incredible!</a><br /></span><p></p><p class="MsoNormal"><span style="font-family: helvetica;"> </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b><i><u>T1B, or Time to 1st Base, is an ideal stat.</u></i></b></span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B is measurable</b>. One camera tracks when the ball is
hit, and another tracks when first base is reached. The difference is T1B. There's
already cameras trained on both locations, so tracking is 'free' in terms of
equipment. </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B is correlated with success. </b>There are many times
when a short T1B make the difference between an out and a single, and there is
no such thing as having a too-short T1B.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B is player-attributable</b>: Nothing the pitcher,
catcher, fielders, or anyone else but the runner themselves has a meaningful
effect on T1B.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B is actionable</b>. Players can improve their times
through sprinting training and by incorporating it into their batting practice.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B cannot be gamed.</b> There are two ways to shorten
T1B – get to base early and hit the ball later. Hitting the ball later earns to
a couple of milliseconds at best.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B has no easy counterstrategy</b>. The only way to
counter a good runner T1B (or RT1B) is with a good fielder time to 1st base
(FT1B). That's going to put pressure on the shift, as well as force errors from
rushed fielding.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"> </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">There are some composite stats like wins above replacement
(WAR) and American football's passer rating that check most or all of these boxes.
But few of them are also as good for fans and media as T1B, because unlike WAR.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B is defined absolutely.</b> WAR and passer rating are subject
to changing definitions based on changes from year to year as well as
differences in opinion.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B is easy to understand and discuss.</b> Is a WAR of
4.0 good? Yes, give me 10 minutes to explain why. Is a T1B of 2.8 seconds good?
Yes, runner go fast.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"><b>T1B is available on the spot.</b> Not only can it be
displayed on the screen right as it happens, broadcasters could even have a
clock ticking upwards like they do for shot clocks in basketball and hang time clocks
in American football.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"> </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">There's an extra benefit to measuring and aiming for a better
average T1B: it prevents players from doing on purpose what 11-year-old me did
by accident – <b>jogging</b>.<br /></span>
<span style="font-family: helvetica;"><br />
An average runner who goes all-out every time is going to have a better T1B than
a fast runner who jogs when they hit a pop-up on even 10-20% of their at-bats.
"<i>Great tacticians learn that consistency often
trumps potential."[2]</i></span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">How often does a player hit a pop fly or something similar and
jog to first base instead of sprinting all-out? Very often. How often is there
an error exactly large enough that running would have made a difference? Not nearly
as often, but more than zero times.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">Arhg! Why are you jogging?! You get 4-5 plate appearances
per game and it's like 30 metres away. Is the risk of injury really that high? Are
you seriously hoping for a double on an error?</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">Just run! Run every time! Run on strike three! Make every at-bat
a fire drill and make fielding, pitching, and catching against you a fearful, stressful
ordeal! Make the game a little more fun to watch! </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">Let's see them shift against that!</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;"> </span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">[1] Dame Ann Marilyn Strathern, summarizing Goodhart's Law.</span></p><span style="font-family: helvetica;">
</span><p class="MsoNormal"><span style="font-family: helvetica;">[2] Resolute Technique, Path of Exile</span></p><span style="font-family: helvetica;">
</span>Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-73869049677515704272020-10-12T14:56:00.001-07:002020-10-12T14:56:30.038-07:00Lost Chapter: Writing for your Career<p class="MsoNormal" style="line-height: normal; margin: 24pt 0cm 3pt; text-align: left;"><span style="font-size: small;"> <span>This is one of the 'lost chapters' of the textbook "<a href="https://www.stats-et-al.com/2020/10/textbook-writing-for-statistics-and.html">Writing for Statistics and Data Science</a>", which was removed because information changes too quickly. This chapter covers data science resumes, describing class projects to businesses, and writing letters of introduction to potential grad supervisors.
<br /></span></span></p><p class="MsoNormal" style="line-height: normal; margin: 24pt 0cm 3pt; text-align: left;"><span style="font-size: small;"><span></span></span></p><a name='more'></a><span style="font-size: small;"><span><br />Part of this has appeared before in <a href="https://www.stats-et-al.com/2017/11/writing-resume-as-data-scientist-or.html">a 2017 post</a>, but a few things have changed since then. For example, advised you to copy/paste a job description
into your resume and to set the text size to 1 and make the words invisible, so
as to get past keyword-searching bots in the resume scanning process. Gaming
the system like that no longer works, so a new section on keywords is included instead.</span></span><p></p><p><span style="font-size: small;"><b><span> </span></b></span></p><p><span style="font-size: small;"><b><span> </span></b></span></p><h3 style="text-align: left;"><span style="font-size: small;"><b><span>Data Science Resumes</span></b><b><span></span></b></span><span style="font-size: small;">
</span></h3><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>About
sending out resumes/CVs/Applications in general.</span></b><span></span></span></p><span style="font-size: small;">
</span><ul style="margin-top: 0cm; text-align: left;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l5 level1 lfo1; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span><span> </span>For specialist jobs like those in data
science, sending a few targeted applications with cover letters is MUCH
better than sending many resumes to everywhere.</span></span></p></li></ul><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><ul style="margin-top: 0cm; text-align: left;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l3 level1 lfo2; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>If you’re
unsure about a job, apply anyways, worry about whether it’s the best for
you, or if you have a chance later. Don’t defeat yourself early.</span></span></p></li></ul><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left; vertical-align: baseline;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left; vertical-align: baseline;"><span style="font-size: small;"><b><span>If you meet all the requirements for a job, you're overqualified.</span></b></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><ul style="margin-top: 0cm; text-align: left;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l1 level1 lfo3; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>Don’t look for
“statistician”, look for “data scientist”, “data analyst”, “business
intelligence”, “business analyst”, “quantitative”, “predictive”.</span></span></p></li></ul><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Consider
the reader:</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Your
audience is NOT another data scientist. It is NON-STATISTICIANS, typically. In
a large company, it will be someone in a human resources department who has
been told to look for certain keywords and skills. They might read your resume
for 20 seconds or less.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>For manual
readers: HIGHLIGHT, or BOLD keywords that show up in the job posting itself.
ESPECIALLY in the QUALIFICATIONS.</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><a href="https://www.monster.ca/jobs/search?q=statistician"><span><span style="color: windowtext;">https://www.monster.ca/jobs/search?q=statistician</span></span></a><span><span> </span>(2 entries in June 2018)</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span><a href="https://www.monster.ca/jobs/search/?q=Data-Scientist"><span style="color: windowtext;">https://www.monster.ca/jobs/search/?q=Data-Scientist</span></a></span><span><span> </span>(162 entries)</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span><a href="https://www.monster.ca/jobs/search/?q=quanitative-analyst"><span style="color: windowtext;">https://www.monster.ca/jobs/search/?q=quanitative-analyst</span></a></span><span>
(6 entries)</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span><a href="https://www.monster.ca/jobs/search/?q=data-analyst"><span style="color: windowtext;">https://www.monster.ca/jobs/search/?q=data-analyst</span></a></span><span>
(312 entries)</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span><br /></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Regarding
keywords. </span></b></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Resumes and
job applications are often scanned by computer for keywords to filter out most
people right away. Some, but not all, of the keywords will appear on job description
itself. If you are applying cold (without being asked to by a friend in the
company), then check the job posting for things that aren't 'generic' like
'data science'.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Look for synonyms
of things that you do. If you talk a lot about prediction, try using the words
"forecasting" or "projection" as well to increase the
chance of getting by the filter with the skills you have.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Research the
company you're applying to. What services do they offer? Try to search for
publicly available resumes of people that might have worked in the target job
before. Often the job description for hiring someone will be made based on how
the previous worker described that job.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>If you are
lucky enough to know someone at the company, ask what words they use to describe
different statistical work, and use those keywords. Don't be afraid to be direct
and ask for the keywords they're looking for.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>There is rarely
a penalty for putting too many potential keywords. </span></b><span>If you're
trying to game the system by sneaking in an entire dictionary of keywords, that
will probably be caught. Otherwise, use plenty.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span><br /></span></b></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span> </span></b></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>In a small
company, your resume may get (a little) more time and may be read by someone
closer to your specialty and (slightly) more familiar with the jargon of your
little corner of your field. The same guiding principle governs both cases:
make it as <b><u>easy as possible for someone to evaluate and say 'yes'.</u></b></span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>What is
this reader going to want to know? “How can this person fill the missing hole
in my organization RIGHT NOW?”</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>This means that
even in highly qualified personnel jobs like those of data scientists, statisticians,
programmers and researchers, that the potential for long term growth within a
company is not a priority, at least not at the resume-reading stage. This is a
major shift from the academic world, where timelines are typically much longer.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>What does
this mean to you, specifically:</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>-
Opportunities come regularly, so <b>don't panic if you don't get the 'right'
position the first time it is posted</b>.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>- Future
plans like the answer to the stereotypical interview question “where do you see
yourself in 5 years” are now irrelevant to many employers. They shouldn't be
mentioned on resume either. <b>Stick to what is solid: The past and present</b>.</span></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>- <b>Emphasize
your skills as they are right now</b>, not where they will be in 6 months.
(e.g. 'I am currently studying...') </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>(</span></b><i><span>as a
corollary, you can say you have skills or qualifications that you are almost
certain to have within a month. It could take that long to start work or get an
interview anyways, and by then it will already be true</span></i><b><span>)</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>- Promising
company loyalty (e.g. 'I have always wanted to work at...') in cover letters
and other correspondence is <b>a waste of time</b> as best, <b>and comes across
as insincere </b>at worst. <b> </b></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>Anybody can
say they are a fan of company xxxx or product yyyy without having any
qualifications. Should my dog work at a meat plant because she loves turkey?</span></i><b><span>)</span></b></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span> </span></b></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto;"><tbody><tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvh7maybTnnjfAHCE13lcvGxKojWijK93mduZSOHx_aHxNXHvoJC9HODaEH2v6wep6XFtOTm0E2ah98Hb_bdfVAv6DOpMXnJDB_FPxtHIkYcu5VEz5E8tgYU0x1bKgIJS6JID5shQkVLxm/s1024/32485954006_3970fd4802_b.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1024" data-original-width="1001" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvh7maybTnnjfAHCE13lcvGxKojWijK93mduZSOHx_aHxNXHvoJC9HODaEH2v6wep6XFtOTm0E2ah98Hb_bdfVAv6DOpMXnJDB_FPxtHIkYcu5VEz5E8tgYU0x1bKgIJS6JID5shQkVLxm/s320/32485954006_3970fd4802_b.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Anyone can present themselves as business ready with the right prep work. [1]</td><td class="tr-caption" style="text-align: center;"><br /></td></tr></tbody></table></span></b></span></p><p><span style="font-size: small;"><b><span> </span></b></span></p><p><span style="font-size: small;"><b><span>On the
subject of transcript grades:</span></b><span></span></span><span style="font-size: small;">
</span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>- After a
certain minimum passing threshold, grades are not a good indicator of job
performance.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>- If you
have graduated, or are on track to graduate soon, then you are already above
this threshold, and no more about your grades needs to be said.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>- The one
exception would be that it is a good idea to mention the awards you have
received related to grades. This includes the dean's list and scholarships.
(e.g. <b>“Graduated in 2017 with distinction”, “Made the Dean's List in 2015”</b>)</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>- Hobbies
and activities from high school are irrelevant unless they are programming
related or you are applying to a position in fast food. In you're reading a
book titled “Writing for Statisticians”, it had better be the first case or you
are severely undervaluing the value of your labour.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>- Rather
than talk about the grades you earn or the courses you took, <b>describe the
projects you did in these courses as experiences</b>. Be specific and clear
without relying on jargon or writing too much, and keep in mind that the reader
is unlikely to be familiar with the course numbers and titles from your
institution.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Example 1:</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Very bad:</span></b><span> “Took Stat
485”</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Bad:</span></b><span> “Took a
course in time-series”</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Good:</span></b><span> “<b>Analyzed</b>
a time-series dataset of the economy of Kansas state.”</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Better:</span></b><span> “<b>Investigated</b>
time-series econometric data, and wrote an executive report.”</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Example 2:</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Bad:</span></b><span> “Took a
course in big data.”</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Good:</span></b><span> “<b>Scraped,
cleaned, and applied</b> a random-forest model to police call data in a Kaggle
competition.”</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Also good:</span></b><span> “<b>Developed</b>
a model to predict crime hotspots from <b>a JSON</b> database from the Seattle
Police Department. Presented findings in a slide deck.” </span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>In each of
the 'good' examples, the experience is written in such a way as to demonstrate
as many high-value skills as possible in a limited space.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>The 'good'
time-series example signals that</span><span></span></span></p><span style="font-size: small;">
</span><ul style="margin-top: 0cm; text-align: left;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l6 level1 lfo4; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You (the
writer) can analyze real data.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l6 level1 lfo4; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You are
familiar with time-series data.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l6 level1 lfo4; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You are
familiar with econometric data.</span></span></p></li></ul><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>The
'better' time-series example signals that</span><span></span></span></p><span style="font-size: small;">
</span><ul style="margin-top: 0cm; text-align: left;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l7 level1 lfo5; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You (the
writer) can analyze real data.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l7 level1 lfo5; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You are
familiar with time-series data.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l7 level1 lfo5; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You are
familiar with econometric data.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l7 level1 lfo5; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You can
communicate your finds to non-specialists.</span></span></p></li></ul><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>The 'good'
big data example communicates that.</span><span></span></span></p><span style="font-size: small;">
</span><ul style="margin-top: 0cm; text-align: left;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l0 level1 lfo6; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You can
analyze big (as in 'high volume') data.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l0 level1 lfo6; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You can scrape
data from the web, or at least an internal database.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l0 level1 lfo6; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You
can prepare and clean data.<span> </span>(50-70%
of most statistical jobs)</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l0 level1 lfo6; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You can format
results into a common government format (i.e. Kaggle).</span></span></p></li></ul><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>The 'also
good' big data example communicates that.</span><span></span></span></p><span style="font-size: small;">
</span><ul style="margin-top: 0cm; text-align: left;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l2 level1 lfo7; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You can
analyze big (as in 'high volume') data.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l2 level1 lfo7; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You can build
predictive, actionable models.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l2 level1 lfo7; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You can work
with JSON data.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l2 level1 lfo7; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>You can
disseminate your findings to non-specialists, such as experts in fields
other than your own.</span></span></p></li></ul><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Use
'business language' to subtly stretch the truth and frame things more
favourably. For example, use the word <b>'setback' instead 'of failure'</b>, or
use <b>'leverage' instead of 'use' or 'exploit'</b>.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Use 'action
verbs' as a helpful guide to demonstrate your experience, especially in the
first work of each statement of your experience. <b>These are verbs that
typically imply leadership, teamwork, or productivity skills. </b>Such words
include, but are not limited to:</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>(<b>distributed,
produced, created, developed, disseminated (i.e. spread), maintained, updated,
cleaned, scraped, prepared, built, wrote, analyzed, coded, investigated</b>)</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>MAKE SURE
YOU KNOW A WORD WELL BEFORE YOU USE IT.</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>In your
experience and even your education, try to start as many sentences as possible
with one of those action words. Remember to write about what you DID and not
what you DO. In other words, use the past tense for everything including your
current position.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>One
apparent exception is when describing duties instead of actions. A popular way
to write about duties instead of actions is to describe duties is to write <b>“responsible
for...”.</b> This sounds like it's present tense, however, it's short for the
past tense<b> “I was responsible for..”,</b> which brings us cleanly to the
next point:</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Taking
advantage of assumptions and formatting:</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>You may
have noticed some things are missing from the 'good' examples of experience.
Specifically, articles and some prepositions are missing. <b>The statements on
a resume should be closer to news headlines than to complete sentences.</b></span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Everything
on a resume is assumed to be about the person whose name is at the top of the
resume. “Statements that start with 'I was' are already longer than necessary;
you and the things you have done are the topics of your resume, so it makes the
most to include key information only, as long as it is not ambiguous. The other
relevant 'what's and 'who's in a good resume are typically made clear from
formatting.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>Consider
the following example:</span></b><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>“Constructed the<span style="background: yellow; mso-highlight: yellow;"></span> database
management system for the company”</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> which can
be shortened to</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>“Constructed
database management system.”</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>while
retaining all or nearly all of the meaning in a resume standpoint. In this
example, the article “the” isn't necessary because without the tests. Likewise,
“for the company” is redundant. </span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Who else
would you be doing this work for, if not the company? (If it was a personal
skill building exercise, you would still leave that information out, the point
is that you have the skill. Why you got it is not important.)</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>The fewer
words you use, while retaining the meaning, the less of those precious 20
seconds of reading time will be to ensure as great as possible a share of that
time is spent observing that you have the qualifications requested in the
document.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 12pt; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Additional
resources:</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span><a href="https://www.monster.com/career-advice/article/sample-resume-computer-programmer-entry-level"><span style="color: windowtext;">https://www.monster.com/career-advice/article/sample-resume-computer-programmer-entry-level</span></a></span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span><a href="https://www.monster.com/career-advice/article/data-scientist-resume-sample"><span style="color: windowtext;">https://www.monster.com/career-advice/article/data-scientist-resume-sample</span></a></span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span><a href="https://resumegenius.com/longest-action-verb-list-universe"><span style="color: windowtext;">https://resumegenius.com/longest-action-verb-list-universe</span></a></span><u><span></span></u></span></p><p style="text-align: left;"><span style="font-size: small;"><span><a href="https://www.glassdoor.ca"><span style="color: windowtext;">https://www.glassdoor.ca</span></a></span><span></span></span>
</p><p class="MsoNormal" style="line-height: normal; margin: 24pt 0cm 6pt; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin: 24pt 0cm 6pt; text-align: left;"><span style="font-size: small;"><b><span>Examples: Data Science Resumes</span></b><b><span></span></b></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>The
following excerpts come from a graduate student's resume before a revision into
a more business friendly format. Take these passages and rewrite them according
the principles of this section. </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>1.</span></b><i><span> “This
thesis develop a diagnostic tool for checking continuous distribution’s
normality, because it is an important assumption to use Integrated Nested
Laplace approximation(INLA)”</span></i></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span> </span></i></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>Possible solution:</span></i></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">1) Developed a diagnostic tool for
checking normality in order to use Integrated Nested Laplace approximation
(INLA) as an replacement to using Markov chain Monte Carlo (MCMC).</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>2. </span></b><i><span>Two main
projects are finished. The first one is about customer behaviours that how to
distinguish a online buyer and a offline buyer, this project is accomplished by
using Hidden Markov Model(HMM). The second project is about push notification
effect, this project is accomplished by three steps. The push notification
effect is defined as click through rate(CTR), it is estimated by two different
approaches, and the best model is chosen by out of sample test.</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Possible
solution:</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Modeled customer behaviours that how
to distinguish online buyers and offline buyers using Hidden Markov Model (HMM)
on click stream data. Investigated the push notification effect on different
conversion metrics.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>3. </span></b><i><span>Detailed
achievements:</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>• </span><i><span>Understand
the company’s database.</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>• </span><i><span>Discover
for right question.</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>• </span><i><span>Solve
questions in a quantitative way:</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>– </span><i><span>Define the
question</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>– </span><i><span>Data clean</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>* Deal with
outliers</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>* Check
data source</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>– </span><i><span>Modeling</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>– </span><i><span>Inference</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>• </span><i><span>How to
present analyzed result to different audiences.</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>• </span><i><span>How to face
failures.</span></i></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span> </span></i></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Possible
solution:</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;"> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Other activities:</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Mastery of corporate databases.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Framing of research questions.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Quantitative Problem solving</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Data cleaning and verification</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Modeling and inference</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Presentation of analysis results to
different audiences.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Perseverance in the face of setbacks
and difficulty.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>4. </span></b><i><span>My main job
is to assistant Postdoctor [name], Associate Research Fellow [name], and
Research Associate [name] to do extensional research on [research or grant
name]. At the same time, I assist the students in the study group on choosing
statistic methods.</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Possible
solution:</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Assisted Postdoctor [name], Associate
Research Fellow Dong Xu, and Research Associate [name] to do extensional
research on ''Application of constructed wetland for water pollution control in
China during 1990–2010''. Assisted students in choosing statistical methods.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;"> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>5. </span></b><i><span>The TA is
response for helping the student in first and second year statistical courses,
and marked the midterms and final exams of these courses. In addition, I marked
the assignments for two statistical courses</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Possible
solution:</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Face to face assistance to students in
first- and second-year statistical courses, grading of assignments and exams.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;"> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>6. </span></b><i><span>This case
competition needs to solve five questions by using the car sensor data. Our
team(3 people) aims to finish three questions, and I finish the question that
distinguish that a car is driving in city or highway. Here is the showcase
Shiny App (link).</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Possible
solution:</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Worked in a team to answer questions
using car sensor data. My primary contribution was distinguishing when a car
was driving in city or on a highway. Created an interactive showcase of results
as a Shiny App (Click to see)</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;"> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><b><span>7. </span></b><i><span>This
project is about to design the best way to travel the breweries in the lower
mainland. The approach is to optimize a function which consider the traveling
distance and the rate of beers.</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Possible
solution:</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="text-align: left;"><span style="font-size: small;"><span style="line-height: 107%;">Designed a method to find the optimal
way to travel the breweries in the lower mainland based on traveling distance
and customer ratings of beers.</span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin: 24pt 0cm 6pt; text-align: left;"><span style="font-size: small;"><b><span>Letters of Introduction (e.g. to grad school supervisors)</span></b><b><span></span></b></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>This
section is about making a cold contact through a letter of introduction. That
is, an introduction of yourself without being prompted or solicited to from
someone else. Being able to do this will greatly improve your networking
prospects.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Start with
a 'My name is... I am a [job / student]” paragraph.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>For clarity
use the phrasing 'My name is [name]' and not 'I am [name]'.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Immediately
after the 'my name is' paragraph, make it clear why you are writing the letter
and what you want.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>One common
goal of a cold introduction is to get an 'informational interview'. In such an
interview, the goal is not to see if you're a good fit for a certain job
position, but with the company, school department, or organization as a whole.
This is your opportunity to learn things about the organization that wouldn't
be easy to find by searching online. See the footnote at the end of this
section.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>A letter of
introduction is NOT a cover letter. A cover letter is specific to a job or
position, whereas this is much more general. A letter of introduction is not a
reference letter either, for similar reasons.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Your letter
of introduction should demonstrate that</span><span></span></span></p><span style="font-size: small;">
</span><ul style="margin-top: 0cm; text-align: left;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l4 level1 lfo8; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>
You’ve done the research to see what this prof’s work is about.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l4 level1 lfo8; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>
You have a specific goal.</span></span></p></li><li class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; mso-list: l4 level1 lfo8; tab-stops: list 36.0pt; vertical-align: baseline;"><p><span style="font-size: small;"><span>
You’ve put time into this introduction (instead of just mass-sending many
similar introductions).</span></span></p></li></ul><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Letter of
introduction template.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>Dear
[person, be sure to use the proper honorific (e.g. Mrs, Mr, Dr). If you are
unsure if a professor has a PhD or not, use 'professor']</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span><br /></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>My name is [your
name], and I am currently [your defining position right now, such as a 4</span></i><i><sup><span>th</span></sup></i><i><span> year
undergrad].</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>I am
interested in [your goal, such as doing a master's degree or getting a position
in financial analysis]*. <b>[Other thing that you did that led you to have this
goal]. [I am interested in doing what you're doing or similar (job), or
interested in doing research in your area, etc.]**.</b> [Mention of plan for
this relationship]*** I would appreciate hearing from you about this line of
work / research (prompting the contact to talk about themselves or their own
work, which people typically enjoy doing)</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>I have included
my resume / CV for review (note that you are not assuming this is for a
particular position, but just in case the contact would like to have it on
hand, or know more about you). If you have time to meet with me, [ways that I
can be contacted].</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>Thank you
for your time!</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><i><span>[Your name
again]</span></i><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>* and **
These two parts are the most important.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>**If you
are trying make a contact with a particular company, you would talk about some
previous work experience that was similar or comparable to what you are trying
to get into.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>*If you are
trying to get into graduate school, this is where you would mention what you
want to study in your program. Be as specific as possible, but don't try to use
terms or jargon that you don't understand – using fancy terms won't impress
anyone. A good example would be “I want to attend graduate school in applied
statistics with a focus on neural networks.”</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>**If you
are trying to get into graduate school, this is where you would mention a
previous course or project that you did that led you to this contact as
potential supervisor. Here you can be VERY SPECIFIC. If it was a class, give
the name of the class and the name of the professor that taught it (there's a
chance that your potential supervisor knows of the professor). If it was an
article or book, mention the title, ESPECIALLY if it was something the
potential supervisor wrote.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>*** For a
company, this would be when you would ask for an informational interview with
“I would appreciate hearing from you about this line of work / research”. This
is an invitation to your contact to talk about themselves or their own work,
which people typically enjoy doing.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>*** For a
potential supervisor, here is where you would VERY briefly describe the kind of
research project you intend to you in your program. This is NOT a commitment to
work on that project, but it does show that you have at least thought about the
research you would be doing and have made a plan.</span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>For more
details for industry, see</span><span><a href="https://www.thebalance.com/letter-of-introduction-examples-and-writing-tips-2062593"><span style="color: windowtext;"> https://www.thebalance.com/letter-of-introduction-examples-and-writing-tips-2062593</span></a></span><span> By</span><span><a href="https://www.thebalance.com/alison-doyle-2058389"><span style="color: windowtext;"> Alison Doyle</span></a></span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>For more
details for academia, see</span><span><a href="https://theprofessorisin.com/2011/07/25/how-to-write-an-email-to-a-potential-ph-d-advisor/"><span style="color: windowtext;"> https://theprofessorisin.com/2011/07/25/how-to-write-an-email-to-a-potential-ph-d-advisor/</span></a></span><span> By </span><span></span></span><span style="font-size: small;"><span></span><span>Dr. Karen
Kelsky</span><span></span></span>
</p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>For further
interest, see:</span><span><a href="https://www.mcgill.ca/gradapplicants/research-supervision/connecting-supervisor"><span style="color: windowtext;"> https://www.mcgill.ca/gradapplicants/research-supervision/connecting-supervisor</span></a></span><span></span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span> </span></span></p><span style="font-size: small;">
</span><p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: left;"><span style="font-size: small;"><span>Footnote:
As a personal example, in an informational interview with a current graduate
student when I was applying to graduate school, I found that the worst thing
about the department I was applying to was the rainy weather in winter. In
other words, there were no academic issues with the department worth
mentioning. Without talking to someone in the department directly, I would have
had to rely of the word of people speaking in a public forum, where complaints
could be discouraged. In private interviews, people tend to be a lot more
candid and honest.<br /><br />[1]</span></span> "My Minion Little Boo"
<span data-v-3d26efba=""> by DaPuglet </span>
is licensed with
CC BY-SA 2.0. To
view a copy of this license, visit
https://creativecommons.org/licenses/by-sa/2.0/
</p><span style="font-size: small;">
</span><span style="font-size: small;"><b><span></span></b></span><span style="font-size: small;">
</span>Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-28591986337384669232020-10-12T12:31:00.007-07:002020-10-12T14:57:51.989-07:00Textbook: Writing for Statistics and Data Science<p>If you are looking for my textbook<br /></p><h2 style="text-align: left;"><a href=" http://www.oercommons.org/courses/writing-for-statistics-and-data-science/view" target="_blank">Writing for Statistics and Data Science</a><br /></h2><p>here it is for free in the Open Educational Resource Commons. </p><p>Writing for Statistics and Data Science is given out under the Creative
Commons 3.0 - Attribution license. That means you and anyone else has
the right to copy it, change it, even sell your version of it, as long
as credit for the original continues to be attributed to me. In short,
it's open source, have fun. </p><p>There were a few chapters that I didn't include because they were either too niche or too prone to becoming obsolete. I'll be posting them here on the blog, with links being added in this post as those chapters go up. Details after the break.<br /></p><p> </p><p><span></span></p><a name='more'></a><p><br /></p><p> If you're looking at this as a distance resource, I've tried to make the notes as self contained as possible, so that someone can just read them, instead of having an abridged version that is filled in with explanations in class. If you are looking as this as a live, post-pandemic resource, this is a course that works well with small group projects at the end of every lecture, especially two-hour lectures where people would otherwise just nod off after 80 minutes. There are in-class exercises for this at the end of some chapters. This book is a collection of the course material that I tried and which, from my limited perspective, produced the best pedagogical outcomes. <br /><br />This is different from other books on composition in that most of it is focused on how to perform specific tasks that someone just finishing a BSc in Statistics or Data Science might be expected to do. Think of it more of a recipe book than a treatise on cooking principles. There are a few early chapters on grammar, style, tone, and audience considerations. (The 'articles: a vs the' chapter is included solely because missing articles is by far the most common grammar error I find when copyediting research papers.) The rest of the book contains guides on how to create questionnaires, poster presentations, technical reports, data dictionaries, and software documentation.<br /><br />Also, as with everything else, it is written from and for a Canadian audience. Normally in stats this wouldn't matter so much; here it does. Your mileage may vary.</p><p>Here are the chapters that are included in the book: </p><ul style="text-align: left;"><li>Digital Writing Tools</li><li>Articles: the vs a</li><li>Tone and Style</li><li>Same Research, Two Audiences</li><li>Equations and Language</li><li>Citing and Intellectual Property</li><li>Reporting Results </li><li>Paper Types and Structure </li><li>Industrial Writing </li><li>Verbal Presentations</li><li>Posters Survey </li><li>Writing Manuals and Instructions </li><li>R Documentation </li><li>Essay Project </li><li>Data Dictionaries</li><li>Intro to Consulting</li><li>Exercise -Scientific Questions</li><li>Exercise -Equations and Language</li><li>Reading Assignment Archive</li><li>Selected Answers</li><li>Abridged Statistical Thesaurus</li><li>Works Cited</li></ul><p><br /><br />The removed chapters, some of which will appear here are:<br /></p><ul style="text-align: left;"><li><a href="https://www.stats-et-al.com/2020/10/lost-chapter-writing-for-your-career.html">Resumes, Describing Class Projects for Jobs, and Letters to Grad Supervisors </a><br /></li><li>Making graphs, </li><li>copyediting, </li><li>making whitepapers, </li><li>additional grammar, </li><li>asking for reference letters.</li></ul><p><br />These aren't in the book because they either relied too heavily on external sources, or because they become out-of-date too quickly, so I'd rather leave them here on the blog where I can remove or update things as needed.<br /><br /></p><p>Here is a personal mirror to the book, but for metrics purposes, I'd prefer if you used the OER Commons link if it's available. <a href="http://factotumjack.com/StatWrit/StatWrit2020.pdf">http://factotumjack.com/StatWrit/StatWrit2020.pdf</a><br /></p><p><br />Thanks to my wife Gabriela for putting up with this side project all these weekends, and apologies to the students of those first few offerings of the course; I hope your other classes were better.<br /><br />-Dr. Jack Davis, PhD, A.Stat<br /> </p><p></p><p></p><p><span face="sans-serif" style="font-size: 22.6px; left: 304.433px; top: 1127.22px; transform: scaleX(0.882867);"></span></p>Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-77031143405703184642020-09-30T19:28:00.003-07:002021-01-21T18:03:57.583-08:00Review of The Theory of Gambling and Statistical Logic<p> <!--[if gte mso 9]><xml>
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<w:LsdException Locked="false" Priority="9" SemiHidden="true"
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<w:LsdException Locked="false" Priority="9" SemiHidden="true"
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<w:LsdException Locked="false" Priority="9" SemiHidden="true"
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<w:LsdException Locked="false" Priority="9" SemiHidden="true"
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<w:LsdException Locked="false" Priority="9" SemiHidden="true"
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<w:LsdException Locked="false" Priority="9" SemiHidden="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="35" SemiHidden="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
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<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
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<w:LsdException Locked="false" Priority="30" QFormat="true"
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<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
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<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
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<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
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<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
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<w:LsdException Locked="false" Priority="21" QFormat="true"
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<w:LsdException Locked="false" Priority="31" QFormat="true"
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<w:LsdException Locked="false" Priority="32" QFormat="true"
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<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
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</p><p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">There are two reasons why I read</span> <span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Review of The Theory of Gambling and
Statistical Logic, Second Edition (2009), by Richard A. Epstein, which dictated
which of the text's 440 pages I paid attention to and which I skimmed.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">First, to learn more of the fundamentals of
betting strategy for my current job at Sportlogiq. Second, to get material to
include in a <a href="https://www.stats-et-al.com/2020/07/statistics-gambling-and-games-of-chance.html">possible future Statistics and Gambling course</a>.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><br /></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span></span></p><a name='more'></a><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><br /></span><p></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Why this book and not something else?</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Other material on the topic like the book
"Weighing the Odds in Sports Betting" by King Yao cover a lot of the
practicality of working with the casinos and some of the ways to find an edge
in a couple particular sports and markets. The podcasts "Guys & Bets
Sports Betting" by Oddsshark, and "Hot Takedown" by
Fivethirtyeight while entertaining are too specific and time-sensitive to learn
the science of smart gambling. Research papers have the theoretical detachment
and staying power to get useful insights, but they're very specific. Papers
give you the right answers; books give you the right questions.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">I won't reveal here much of what I learned for
work, but I'll share all my notes on the teaching possibilities.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p><p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDfH7bkhVvpcFBue40V76x8qbqo8AfnL-tq3FL4F0z4pFdYUEiZtuhRIUWqE8w55z1SqzJCVRbbFlz4tunHUnnpUjqASeNGxIdDVrXspiajzHcotDRXavK6uZTaIZPlxZTtVnkFnLCIKdg/s616/Theory+of+Gambling+stats-et-al+com.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="616" data-original-width="415" height="375" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDfH7bkhVvpcFBue40V76x8qbqo8AfnL-tq3FL4F0z4pFdYUEiZtuhRIUWqE8w55z1SqzJCVRbbFlz4tunHUnnpUjqASeNGxIdDVrXspiajzHcotDRXavK6uZTaIZPlxZTtVnkFnLCIKdg/w253-h375/Theory+of+Gambling+stats-et-al+com.jpg" width="253" /></a></div><br /> <p></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The Parrando Principle</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The Parrando Principle is basically this: You
can turn two losing games into a winning game if you are allowed to choose
which rule set you're going to use at each step of the game. A winning strategy
might even be to randomly choose between the rule sets.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">According to the book, it's a theoretical
breakthrough and the next step beyond minimax strategies. Aside from a
stock-trading strategy, I couldn't see the practicality of it, though. Most of
the examples were these contrived mathematical novelties, but pay attention to
this idea in this future.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The Epaminondas system</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The Epaminondas system (examples given in the
roulette and craps sections) for detecting gambling edges (say, from a worn out
roulette wheel or imperfect dice) uses extreme values. It needs less data to
reach statistical significance than the more classical chi-squared test, which
tests the entire observed distribution against a theoretical one.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Utility Curves</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">page 46-47 talks about utility curves, and
describes a person's utility curve with a concave component for small gains and
losses, and a more traditional convex component for large gains and losses. In
other words, winning $10 helps more than losing $10 hurts, but winning $100,000
helps a lot LESS than losing $100,000 hurts.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The convex utility curve is used a lot in
actuarial work to explain why people purchase insurance, even though on average
it costs more than it pays out - some financial losses are too large to recover
from.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">This version of the utility curve also factors
in the small emotional thrill of gambling, I suppose. In other words, losing a
small gamble is just paying for entertainment, while winning a small gamble is
profiting and getting entertainment.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The amount of concavity is called 'the
acceptable amount of unfairness'.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Wagering and Bankroll management</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Pages 60-65 are about bankroll management,
including a mathematical derivation of the Kelly Stakes Criterion, a surprisingly
simple system to implement that provably maximizes your profit relative to you
bankroll. The catch is that you need to make profitable bets, and you need to
know the probability of an event happening. These profitable opportunities in
"real-world" events like sports, races, and elections where the true
probability is unknown by the market. The Kelly Stakes criterion doesn't tell
you what to bet on, but it does tell you how much to bet, assuming you have an
'edge', where you know the probability better than the market. </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">This book covers an adjustment to Kelly based
on your confidence in your probability estimate. The book also talks about
other criteria like practical limitations on the number of wagers you can make.
</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Besides limited information or number of bets,
there are many other modifications to Kelly besides those in the book. If one
or more bets is going to take a substantial time to pay out, then a
Simultaneous Kelly strategy makes sense. If you are using additional
risk-management strategies like hedging, then you can wager more than Kelly
would suggest on the opening part of a bet.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Bingo</span></b><b><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"></span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Page 155 states that there are (15 choose 4)
or 1365 "distinct" bingo cards because that's the number of possible
center columns. It also states there are (15 choose 4) X (15 choose 5)^4
"possible" bingo cards. I don't know what the author means by
"distinct" here, and a quick internet search came up with no answer.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The maximum possible number of cards that be
played together that can produce exactly 1 winner is 15^4. Any more than that
and you're guaranteed to have more than 1 winning card at the same time,
whenever there is a winner.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Page 156 has a table of the expected number of
numbers called before there is a bingo, which I've done for one card. However,
the author provides the time-to-bingo for N < 15^4 simultaneous cards, which
is more useful if you're planning a game. It would make a cool simulation
project.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">On page 157, there is a quick proof that in
multi-card games, if there are multiple winners in a game, that it's more
likely to be wins along the column instead of along the rows. It's true, but I
would have never suspected without thinking it through.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">What's even weirder is that if there's any
bingo at all, it's about twice as likely to come from a row.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">So column wins come all at once, but row wins
are more frequent. Weird.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Dice Games</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Pages 174 and 175 talk about De Mere's
Problem: How many dice rolls are needed until some arbitrary outcome happens?</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">It's a geometric distribution, and the minimum
number of rolls n until the outcome happens with probability p is a cute
formula:</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">pn = probability of outcome happening in the
first n rolls.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">p1 = probability of outcome happening in any
given roll.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span lang="PT-BR" style="font-size: 12.0pt; line-height: 107%; mso-ansi-language: PT-BR; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">n = - log(1 - pn) /
log(1 - p)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span lang="PT-BR" style="font-size: 12.0pt; line-height: 107%; mso-ansi-language: PT-BR; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">so it takes...</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">n = log(1 - 1/2) / log(1 - 1/6) = log(2) /
(log(6) - log(5)) = 3.802</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">rolls of a 6-sided die until there's a 50%
chance of it rolls on 6. It's not complicated, but it's cute and I like it.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Page 183 features James Bernoulli's game:</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Step 1) Roll a die, call the result x.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Step 2) Roll x dice. Your score is the total.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">If the total is more than 12, you win $1.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">If the total is 12, you won/lose $0.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">If the total is 11 or fewer, you lose $1.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">What's unusual about this game is that
E[score] = 12.25, but E[payout] < 0.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Specifically, Pr[score > 12] = 0.468,
Pr[score < 12] = 0.484.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">This is a good lesson on how over/unders and
point spreads should be built around the median outcome, not the mean outcome.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">It's not in the book, but high-variance
strategies when losing in sports work on a similar principle. When a hockey
team that's behind by 1 pulls their goalie, the average number of goals they
will lose by increases, but so does their chance of winning.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Page 184 talks about the optimal solution for
a 'maximum roll' game that seems like a proto-Yahtzee. In this game, five dice
are rolled, and at least one die is kept or 'frozen', and the rest are
re-rolled. This process is repeated for the rolled dice until every die is
'frozen'.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The goal is get the highest average sum.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The optimal strategy can be found with the
dynamic programming approach summarized below:</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">If you roll two dice, you can freeze either
one or both of the dice. Since the expected score from rolling a d6 is 3.5, the
optimal strategy is to keep both when both are in {4,5,6}, and to re-roll the
lower die otherwise. This strategy produces an expected score of 8.236.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">If you roll three dice, you can freeze one,
two, or all three.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">If the sum of the lowest two of the three are less
than 8.236, reroll those two and use the two-dice strategy. Otherwise, if the
lowest one is less than 3.5, reroll that one. This strategy produces an
expected score of 13.425.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A similar strategy produces an expected score
of 18.844 for 4 dice, and 24.442 for 5 dice.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Only two pages, 193 and 194 were written on
Yahtzee. There's a handy table on the expected value of aiming for different
scoring possibilities, which I assume were derived from a similar dynamic
programming method as the maximum dice in the previous example. I'm still
confused about the possibility of changing your goal after 1 or 2 rolls if
there is unexpected results.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Page 195 talks about non-transitive dice, or
rock-paper-scissors dice. Essentially that you can assign scores to different
faces on dice such that E[score A] = E[score B] = E[score C], but the
Pr(A>B), Pr(B>C), Pr(C>A) are all more than 0.5. In short, dice A, B,
and C act like rock, paper, and scissors in a competition to roll higher than
your opponent.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Pages 200-205 talk about Craps, but it's just
a game description.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Page 207 talks about the average duration of a
craps hand, which is a little more system.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Page 209 proposes a game of craps with 8-sided
dice. That's a good idea for problems and exercises in general - how do
classical games change when there are dice with more sides, or unfair dice?</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">page 212-213 talks briefly about backgammon,
specifically how AI has quietly become a superhuman force in the game. What
makes this interesting is the random element of rolling to move; a
minimax-based search just isn't feasible like it is with chess.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Exercise from book: Prove no weighting of two
6-sided dice can make Pr(sum = 2) = Pr(sum = 3) = ... = Pr(sum = 12)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Possible expansion: Is this true for n dice or
d sides?</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Card Games in General</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Around page 210 there is a discussion of card
shuffling methods and the amount of entropy they produce. </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">For example, a perfect shuffle produces
log_2(52!) bits of entropy because each of the 52! deck orders is equally
likely. Riffle shuffling and even bridge shuffling (interleaving) produces much
less entropy. </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">As another example, an ideal cut produces
log_2(52) bits of entropy because there are 52 equally likely configurations
that the deck can be in. In reality, cuts closer to the middle of the deck are
more likely, so the entropy is lower.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">In a course, card decks is a good place to
talk about sampling with out without replacement.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The book talks briefly about a game called
"roller coaster dice" where a player rolls 2d6 and then predicts if
the next 2d6 roll will be higher or lower than the previous roll. A similar
game can be played by drawing cards from a deck, and prediction can be improved
by keeping track of the cards that have already been seen and 'removed' from
the deck.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A useful programming project to demonstrate
the advantage gained from accounting for replacement could be to simulate
playing this game using two strategies and tracking the average number of
successful guesses in a deck (in total or until first failure).</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">"With replacement" strategy of
guessing 'greater' after drawing 2-7, 'less' after drawing 9-A, and either one
after drawing an 8.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">"Without replacement" strategy of
guessing 'greater' when more than half the remaining cards have a greater rank
than a previous card, and 'less' otherwise.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">page 229 has some information on the expected
number of cards needed to draw until a certain combination is encountered,
which is useful for trick-taking games like Bridge, Canasta and Shanghai, or
collectible card games like Magic: The Gathering.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Poker</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">pages 248-260 is on poker both in general and
Texas Hold-Em.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">There are thousands and thousands of resources
specifically for poker, so I only took down a few tables and references of
interest.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The probability of various hands for a 5-card
hand (e.g. for draw poker), the best 5 of a 7-card hand (e.g. Texas Hold-Em),
and the multiplier between them (which I added).</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Hand</span></p>
</td>
<td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Pr(in 5 cards)</span></p>
</td>
<td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Pr(In 7 cards)</span></p>
</td>
<td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Multiplier</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">High Card</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.5012</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.1740</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">0.347</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">1 Pair<span style="mso-tab-count: 1;"> </span></span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.4226</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.4380</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">1.036</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">2 Pair</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0475</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.2350</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">4.947</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">3 of a kind</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0211</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0483</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">2.289</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 5;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 6;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Straight</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.00392</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0462</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">11.79</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 7;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Flush</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.00199</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0303</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">15.23</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 8;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Full House</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.00144</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0260</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">18.06</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 9;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">4 of a kind</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.00024</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.00168</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">7.00</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 10;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 11;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Straight Flush</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.000014</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.000311</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">22.21</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 12; mso-yfti-lastrow: yes;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Royal Flush</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.85pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0000015</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.000033</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 116.9pt;" valign="top" width="156">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">22.21</span></p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-tab-count: 3;"> </span></span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">There was also a table on 3-card hand
probabilities, which are useful in games like Texas Hold-Em for the 3 cards of
the flop.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.8pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Hand</span></p>
</td>
<td style="border-left: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.85pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Probability</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.8pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">High Card</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.85pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.828</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.8pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">1 Pair</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.85pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.169</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.8pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">3-Flush</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.85pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0518</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.8pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">3-Straight</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.85pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.0347</span></p>
</td>
</tr>
<tr style="mso-yfti-irow: 5; mso-yfti-lastrow: yes;">
<td style="border-top: none; border: solid windowtext 1.0pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.8pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">3 of a kind</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 155.85pt;" valign="top" width="208">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">.00235</span></p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">According to Shackleford and Galfond, the
following hands are worth playing pre-flop (when all you have are the two cards
in front you and no community cards) if everyone ahead of you has folded
already.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Early Position: </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">77 88 99 TT JJ QQ KK AA</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">AT AJ AQ AK</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">KTs KJ KQ</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">QTs QJs T9s</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Which is the top 12.8% of hands</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">(Note T stands for 'ten', and s stands for
'same suit'). </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">In late position, since you have fewer players
left to play, the playable range extends to also include:</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A6s A7s A8s A9s</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">K8s K9s KT</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Q8s Q9s QT QJ</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">J8s J9s JT</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">T8s</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">98s</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Which is the top 25.5% of hands.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">page 268 - Blackjack Basic Strategy.</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Possible Homework/Lecture project - How does
the basic strategy change when the count is high (i.e. there are relatively
more high cards left to be played in the shoe/deck.), or when there are
multiple decks? What if there we removed all the hearts and diamonds?</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">page 271 - Card Counting in Blackjack.</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Peter Griffin has a concept called EOR (Effect
of Removal), which is essentially the amount that the game gets worse for the
dealer (and better for you) when certain cards are removed from the deck.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">When the total EOR is positive, the dealer is
more likely to bust than average, and therefore the expected value of your bet
improves (although rarely will it improve to positive)</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 1184;">
<tbody><tr style="height: 12.95pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Rank</span></p>
</td>
<td style="border-left: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Effect of Removal</span></p>
</td>
</tr>
<tr style="height: 12.95pt; mso-yfti-irow: 1;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">2</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">0.381</span></p>
</td>
</tr>
<tr style="height: 13.65pt; mso-yfti-irow: 2;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 13.65pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">3</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 13.65pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">0.434</span></p>
</td>
</tr>
<tr style="height: 12.95pt; mso-yfti-irow: 3;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">4</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">0.568</span></p>
</td>
</tr>
<tr style="height: 12.95pt; mso-yfti-irow: 4;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">5</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">0.727</span></p>
</td>
</tr>
<tr style="height: 12.95pt; mso-yfti-irow: 5;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">6</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">0.412</span></p>
</td>
</tr>
<tr style="height: 12.95pt; mso-yfti-irow: 6;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">7</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">0.282</span></p>
</td>
</tr>
<tr style="height: 13.65pt; mso-yfti-irow: 7;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 13.65pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">8</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 13.65pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">-0.003</span></p>
</td>
</tr>
<tr style="height: 12.95pt; mso-yfti-irow: 8;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">9</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">-0.173</span></p>
</td>
</tr>
<tr style="height: 12.95pt; mso-yfti-irow: 9;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">10</span></p>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border-left: none; border-right: solid windowtext 1.0pt; border-top: none; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 113.6pt;" valign="top" width="151">
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">-0.512</span></p>
</td>
</tr>
<tr style="height: 12.95pt; mso-yfti-irow: 10;">
<td style="border-top: none; border: solid windowtext 1.0pt; height: 12.95pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 56.45pt;" valign="top" width="75">
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<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Q</span></p>
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<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">-0.512</span></p>
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<p class="MsoNormal" style="line-height: normal; margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">-0.579</span></p>
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<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">On page 274, there are simple point-count
systems to approximate this, such as the Hi-Lo count that the MIT Blackjack
team used.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">If you're thinking about going to a casino
blackjack table to make a lot of money counting cards, consider something that
will net you more money with less work, like making YouTube videos, or standing
alone in a parking lot.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Horse Racing, Parimutuel</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Page 300 talks about the parimutuel, or 'bet
among yourselves' system used in horse racing. In a parimutuel system, the odds
offered at any given point (except right after the opening odds have been
offered) are based on the proportion of money backing that horse or outcome.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The formula is </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">(decimal odds) = (1 - vig) / (proportion of
bet money on this outcome).</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The vig, or proportion of bets that the
racetrack takes, is typically higher in parimutuel than it is in sports. A
typical quoted amount was 12%, so a $1 bet on a horse that had 20% of the money
behind it would pay out</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">$1 x (1 - 0.12) / (0.20) = $4.40</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">As the proportion of money changes, these odds
can change a great deal too. Like other sports gambling, the odds you're quoted
on are the odds your winning are based on, not the closing odds. That
volatility and the maintenance of the horse track might justify the higher vig.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><b><span style="font-size: 14.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">References worth a follow-up:</span></b></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Other older games like Primero, Trijaques,
Quinquenoue, Hazard, Esperance, Troid Dez, Passedix, Rafle, Her Jeu des Tas.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">David Agard and Michael Shackleford - "A
New Look at the Probabilities in Bingo," The College Mathematics J. 33:4,
(2002) 301-305.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Stewart N. Ethier - Optimal Play: Mathematical
Studies of Games and Gambling.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Peter Griffin - The Theory of Blackjack</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Phil Galfond - "G-Bucks Conceptualizing
Money Matters" April 2007 - Bluff Magazine.community</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Martin Gardner - Mathematical Carnival</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">John D. Beasley: The Mathematics of Games.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Michael Shackleford: "Texas Hold-Em"
The Wizard of Odds.</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">King Yao: "Weighing the Odds in
Poker".</span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span></p>
<p class="MsoNormal" style="margin-bottom: 0cm; text-align: justify;"><span style="font-size: 12.0pt; line-height: 107%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">Phil Woodward: "Yahtzee: The
Solution" Chance 16:1 (2002) 17-20.</span></p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-65301293902541393952020-08-22T23:26:00.001-07:002020-08-22T23:26:29.216-07:00Fantasy League Sports Cards<p> <!--[if gte mso 9]><xml>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 5"/>
<w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Closing"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Signature"/>
<w:LsdException Locked="false" Priority="1" SemiHidden="true"
UnhideWhenUsed="true" Name="Default Paragraph Font"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Message Header"/>
<w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Salutation"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Date"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Block Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 1"/>
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</p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">The sports
card industry (specifically baseball cards) crashed in 1994. Fantasy sports existed
as early as the 60's, but really caught public attention around 1995. That timing
is not coincident.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">In both hobbies,
fans get to have surrogate ownership of players, and the market value
of those surrogates goes up or down with the performance of
those players. At the casual level, being in a fantasy league is just a more publicly
acceptable way to collect and play with cards. At the serious level, fantasy is a
more viable, faster way to make a profit with your expertise than cards were.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">In short, fantasy
is just trading cards for grown ups.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">But what if physical cards let you draft players?</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> <span></span></span></p><a name='more'></a><p></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Imagine each
card had a unique QR code; scanning that code with an app would add that
player to your available draft pool for fantasy competitions on that app. </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">If someone
else scans that code, the player is added to their draft pool and removed from
yours. That way, the number of a player available is limited by the number of copies
of the card that are printed.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">If the company
printed 10,000 copies of Shohei Ohtani, then you would have 10,000 different
QRs, and 10,000 Ohtanae available in a massive fantasy pool. Each card might look like this:<br /></span></p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgti8LVGfaIS3vfAD11qCmU3bSzEs160PuJDaRcEMXNSSk3p53LivlSQtYs6cvyDYBvH7lKLSQw3BtkLSO_c0F76lHNgMB84MKuLteIW9IvQRM_1Z1r2VT_m_NY9K73aWzutNu6EKSTyHVF/s1398/fantasy+cards.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1164" data-original-width="1398" height="426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgti8LVGfaIS3vfAD11qCmU3bSzEs160PuJDaRcEMXNSSk3p53LivlSQtYs6cvyDYBvH7lKLSQw3BtkLSO_c0F76lHNgMB84MKuLteIW9IvQRM_1Z1r2VT_m_NY9K73aWzutNu6EKSTyHVF/w512-h426/fantasy+cards.png" width="512" /></a></div> <p></p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> <br /></span></p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"><span style="font-size: 12.0pt; line-height: 107%;"><!--[if gte mso 9]><xml>
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Name="Hashtag"/>
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</xml><![endif]--></span>I will now
take questions from the audience.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: Why
would a company ever do this?</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">A card
company could drive card sales by putting up prizes in the fantasy pool.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">A fantasy
company could have an exclusive gimmick that lets them reach fans that purely
digital fantasy companies couldn't.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">It's possibly
a way to get around anti-gambling laws because fans aren't paying money to enter
a fantasy league. Similarly, some leagues can offer future packs of cards as
prizes instead of money. Also, 'no purchase necessary' requirements could be
met by starting every fan with a limited 'starter' pool of players that allows
them to participate but not effectively compete.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: How is
this better for fans? <br />
</span></b><span style="font-size: 12.0pt; line-height: 107%;">It's a value-add to card
collectors. Cards already exist as a way to sell memories, and this gives card collectors
a way to participate and make more memories.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">It's a
value-add to fantasy players. They still have something, possibly of collector
value, after the fantasy league is over.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: How is
this worse for fans?</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Without an (possibly
expensive?) modification to the printing process, <span style="mso-spacerun: yes;"> </span>the codes might need to be on a sticker placed
on the card. That's not good from an aesthetic standpoint.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">You probably
have to handle the cards to scan them, which is bad for serious collectors.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: Does
this make fantasy sports really pay-to-win?</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Possibly. The
fantasy game can become pay-to-win, as top prospects become a scarce, necessary
commodity to compete. </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">One way to
work around this is to print more copies of popular, better players – a common practice
in sports cards, but the opposite of many trading card games.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p><p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"><span style="font-size: 12.0pt; line-height: 107%;"></span>
</span></p><p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: So the
cards become worthless after the player retires?</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Their fantasy
value drops to zero, but their collector value should increase because of the
retirement.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: How am
I supposed to trade players with other fans? </span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Before 2020
we had things like 'meeting people' and 'the post office'. Those things might
return. </span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: Don't be
a pretentious jerk, I live in a small town.</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Sorry. Maybe
send pictures and then proof of destruction of QR codes? There are pin traders in
small towns.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: Won't people
just copy cards?</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">They can
try. Before QR codes, Sony tried something similar with a game called Eye of
Judgement, but each copy of a card had the same code, so players could just
print and copy the cards they wanted to play the game.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Each QR is
unique for that one physical card. <span style="mso-spacerun: yes;"> </span>Also,
the number of possible QRs is much greater than the number of cards that would
be feasibly printed, so generating QR codes to guess at which ones lead to player
drafts isn't feasible either.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: If I
trade a card away, why can't I just take a picture of it first to retain the
player by re-scanning the picture?<br />
</span></b><span style="font-size: 12.0pt; line-height: 107%;">Make each card scannable
unlimited times, but only once per account (and lock account to mailing address,
for cash prizes).</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: If I
trade a card away, why I can't I just get another account somewhere else to
scan all my traded players back?</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">That's a real
problem and I don't have a good answer yet. Anything like this needs a good customer
support team, but that's yet another expense. Hopefully it's detectable.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p>
<p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: What
if I take a picture of a card, then trade away the card, and then later give
the picture to a friend so they can scan it after my victim scans it?</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Okay, but… if
the printed card distributions minimize the pay-to-win aspect, this is a lot of
effort for a limited gain.</span></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;"> </span></p><p class="MsoNormal"><b><span style="font-size: 12.0pt; line-height: 107%;">Q: Isn't
this a digital activity with added physical clutter?</span></b></p>
<p class="MsoNormal"><span style="font-size: 12.0pt; line-height: 107%;">Sort of, but cards aren't
that big, and they're mostly made of paper.<br /></span></p>
Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com1tag:blogger.com,1999:blog-5718736365000160223.post-84009687500657492052020-08-13T18:52:00.005-07:002020-10-30T09:35:10.400-07:00Soccer-to-Hockey Translation GuideHockey and soccer (ice hockey and football) are similar
enough from a fan or analytics perspective that if you’re familiar with one,
it’s easy to become familiar with another. There could be a whole new world of
sport you’re missing out on!
<p class="Standard">In this article I’ve organized many of the parallels and
contrasts between hockey and soccer so that you can watch a few games and of
either one and confidently say that "hockey is just like soccer except X
instead of Y".</p>
<p class="Standard"> </p><span><a name='more'></a></span><p class="Standard"> </p>
<p class="Standard">I’m more familiar with hockey, and while the main intent of
this guide is to introduce fans of soccer to hockey, I hope that it the guide
is helpful in both directions.</p>
<p class="Standard"> </p>
<p class="Standard">Differences that I’ve found that surprize others the most are
in <b><i>bold italics</i></b>. Specifically, that player substitutions in
hockey are far more common, that hockey players can hit each other, and that
there is playable area <b><i>behind </i></b>the net.</p>
<p class="Standard"> </p>
<p class="Standard"> </p>
<p class="Standard"><b>Which version of each game?</b></p>
<p class="Standard"> </p>
<p class="Standard">There are many types soccer and hockey played around the
world. This guide will refer to the game outlined in IFAB’s “The Laws of the
Game” and endorsed by FIFA, as found here: <a href="https://resources.fifa.com/image/upload/ifab-laws-of-the-game-2020-21.pdf?cloudid=d6g1medsi8jrrd3e4imp"><span color="windowtext" style="text-decoration: none; text-underline: none;">https://resources.fifa.com/image/upload/ifab-laws-of-the-game-2020-21.pdf?cloudid=d6g1medsi8jrrd3e4imp</span></a></p>
<p class="Standard"> </p>
<p class="Standard">Likewise, hockey will refer to the game described in “The NHL
Rulebook” found here: <a href="http://www.nhl.com/nhl/en/v3/ext/rules/2018-2019-NHL-rulebook.pdf"><span color="windowtext" style="text-decoration: none; text-underline: none;">http://www.nhl.com/nhl/en/v3/ext/rules/2018-2019-NHL-rulebook.pdf</span></a></p>
<p class="Standard"> </p>
<p class="Standard"><b>Why “soccer” and not “football”? </b>To minimize
confusion. People may scoff when you say ‘soccer’ but they’re never confused.</p>
<p class="Standard"> </p>
<p class="Standard">If I get around to translating NFL/XFL/CFL football to rugby,
I’ll call football ‘gridiron’ or something so that there’s nobody gets to use
‘football’.</p>
<p class="Standard"> </p>
<p class="Standard"><b>Why “hockey” and not “ice hockey”?</b> To the field hockey
fans out there, both of you can send me stern letters at <a href="mailto:jackd@sfu.ca"><span color="windowtext" style="text-decoration: none; text-underline: none;">jackd@sfu.ca</span></a> .</p>
<p class="TableContents"> </p>
<p class="TableContents"> </p>
<p class="TableContents"><b>You should do this for other sports.<br />
</b>I did, for baseball and cricket. <a href="https://www.stats-et-al.com/2018/08/baseball-cricket-translation-guide.html"><span color="windowtext" style="text-decoration: none; text-underline: none;">https://www.stats-et-al.com/2018/08/baseball-cricket-translation-guide.html</span></a></p>
<p class="TableContents"><b> </b></p>
<p class="Standard"> </p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed;">
<tbody><tr style="height: 20.95pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid; height: 20.95pt; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Soccer (Football)</span></b></p>
</td>
<td style="border: 1pt solid; height: 20.95pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Hockey (Ice Hockey)</span></b></p>
</td>
</tr>
<tr style="height: 18.75pt; mso-yfti-irow: 1;">
<td style="border: 1pt solid; height: 18.75pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Playing Area</span></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; height: 18.75pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Playing Area</span></b></p>
</td>
</tr>
<tr style="height: 139.85pt; mso-yfti-irow: 2;">
<td style="border: 1pt solid; height: 139.85pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The field is between 90m and 120m in length and
between 45m and 90m in width.</p>
<p class="TableContents"> </p>
<p class="TableContents">See page 37 of IFAB: Laws of the Game at <a href="https://resources.fifa.com/image/upload/ifab-laws-of-the-game-2020-21.pdf?cloudid=d6g1medsi8jrrd3e4imp"><span color="windowtext" style="text-decoration: none; text-underline: none;">https://resources.fifa.com/image/upload/ifab-laws-of-the-game-2020-21.pdf?cloudid=d6g1medsi8jrrd3e4imp</span></a>
for details.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; height: 139.85pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The rink is 60.96m long and 21.33m wide in North
America.<br />
<br />
The rink is exactly 60m long and 21m wide in the rest of the world.</p>
<p class="TableContents"> </p>
<p class="TableContents">See <a href="http://www.sportsfeelgoodstories.com/hockey-rink-dimensions-size-diagram/"><span color="windowtext" style="text-decoration: none; text-underline: none;">http://www.sportsfeelgoodstories.com/hockey-rink-dimensions-size-diagram/</span></a>
for details.</p>
</td>
</tr>
<tr style="height: 84pt; mso-yfti-irow: 3;">
<td style="border: 1pt solid; height: 84pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The net 732cm wide and 244cm tall.</p>
<p class="TableContents"> </p>
<p class="TableContents"><b><i>The is no playing surface behind the net, so
depth is irrelevant.</i></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; height: 84pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The net is 182.9cm wide, 137.2cm tall, and 101.6cm
deep.</p>
<p class="TableContents"> </p>
<p class="TableContents"><b><i>There is 335cm (NA) or 400cm (World) of playing
surface behind the front of the net.</i></b></p>
</td>
</tr>
<tr style="height: 55.8pt; mso-yfti-irow: 4;">
<td style="border: 1pt solid; height: 55.8pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The playing surface is marked with painted lines, but
there are no barriers to prevent the ball from rolling off of the field.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; height: 55.8pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Fibreglass boards and Plexiglas walls stop the puck
from leaving play. Players often bounce the puck or each other off the boards
intentionally.</p>
</td>
</tr>
<tr style="height: 28.15pt; mso-yfti-irow: 5; mso-yfti-lastrow: yes;">
<td style="border: 1pt solid; height: 28.15pt; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">All markings are painted on top of the grass or turf.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; height: 28.15pt; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">All markings are painted underneath the ice and are
visible through it.</p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4QHEY13_T2Ml5w1DhCuq_xEPhYwFz9OS4pb38mwSY_qTAyUH3Wdb1Y5b8vDf16mFrg-f2bwkOmiZHf6iYJ8EQB8dI_w3FUkMwChS28UUYCYDcrvZeP4ml7r8m_4N6zXH50zsjWnGuASNn/s660/Hockey-Rink-Dimensions-for-NHL-Diagram.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="419" data-original-width="660" height="325" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4QHEY13_T2Ml5w1DhCuq_xEPhYwFz9OS4pb38mwSY_qTAyUH3Wdb1Y5b8vDf16mFrg-f2bwkOmiZHf6iYJ8EQB8dI_w3FUkMwChS28UUYCYDcrvZeP4ml7r8m_4N6zXH50zsjWnGuASNn/w512-h325/Hockey-Rink-Dimensions-for-NHL-Diagram.jpg" width="512" /></a></div><p></p>
<p class="MsoNormal"> </p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed; width: 590px;">
<tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Soccer (Football)</span></b></p>
</td>
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Hockey (Ice Hockey)</span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Flow of Play</span></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Flow of Play</span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Soccer is played over 2 halves of 45 minutes each.</p>
<p class="TableContents"> </p>
<p class="TableContents">The match clock continues when play is stopped, but
some time is added to the end of each half at the referee’s discretion based
on the amount of time lost dealing with some stoppages like injuries.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Hockey is played over 3 periods of 20 minutes each.</p>
<p class="TableContents"> </p>
<p class="TableContents">The game clock is stopped whenever play is stopped.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">In regular season matches, ties after 90 minutes are
resolved as a draw.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">In regular season games, ties after 60 minutes are
resolved with a 5-minute overtime that ends as soon as one team scores a
goal. If the game is still tied, teams resolve the tie with penalty shots.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">In elimination matches, ties after 90 minutes are
resolved first with two halves of 15 minutes each. If the match is still
tied, the tie is resolved with penalty kicks.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">In elimination games, ties are resolved with
additional 20-minute periods until one team scores a goal.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 5;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Some elimination rounds are two matches long, with
each team acting as the home team for one match. In these rounds, the total
number of goals determines the winner. If there is a tie, the team that
scored more goals as the visitor wins. If the round is still tied, single
match tie breakers are used.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Multi-game eliminations are resolved with an odd
number of independent games, usually 7. The first team to win a majority of
the games wins the round.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 6; mso-yfti-lastrow: yes;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Play starts with one team controlling the ball.</p>
<p class="TableContents"> </p>
<p class="TableContents">Play is restarted with a throw-in if it leaves play
from the side, a corner kick if it leaves play from the end by the defending
team, or a goal kick if it leaves play from the end by the attacking team.</p>
<p class="TableContents"> </p>
<p class="TableContents">A throw-in is an over-the-head throw of the ball back
into play by a player from the team that did not touch the ball last when it
left play.</p>
<p class="TableContents"> </p>
<p class="TableContents">A corner kick is a kick into play by the attacking
team.</p>
<p class="TableContents"> </p>
<p class="TableContents">A goal kick is a kick from the goalkeeper’s area by
the defending team.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Play is always started or restarted with a faceoff, in
which the referee drops the puck for one player from each team to contest.</p>
<p class="TableContents"> </p>
<p class="TableContents">At the start of each period or after a goal, the
faceoff is taken from center ice.</p>
<p class="TableContents"> </p>
<p class="TableContents">The location of other faceoffs depends on the last
legal position of the puck before the stoppage of play.</p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0RBZVHBE8fv98i2JPnKSoyjP8U20RgD0cZAnh4cvfxq2asuNi0C4wVbn8pv7WVU5AeVNrBfvqKQVaXxLjsPJ_BB7CL7k-JFcrvgRjpTUo_Nd8bWjE376C456U59LCfCArTdJ4xT_2scg2/s861/Soccer+Field+Dimensions.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="659" data-original-width="861" height="392" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0RBZVHBE8fv98i2JPnKSoyjP8U20RgD0cZAnh4cvfxq2asuNi0C4wVbn8pv7WVU5AeVNrBfvqKQVaXxLjsPJ_BB7CL7k-JFcrvgRjpTUo_Nd8bWjE376C456U59LCfCArTdJ4xT_2scg2/w512-h392/Soccer+Field+Dimensions.PNG" width="512" /></a></div><p></p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed; width: 590px;">
<tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Soccer (Football)</span></b></p>
</td>
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Hockey (Ice Hockey)</span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Equipment</span></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Equipment</span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The ball has a durable flexible surface. It weighs
between 410 and 450 grams, and has a diameter of 22cm. Smaller balls are used
for training and junior leagues.</p>
<p class="TableContents"> </p>
<p class="TableContents">The ball in inflated to between 600 g per cubic m and
1100 g per cubic m of pressure at the beginning of the match.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The puck is a flat, solid disk 2.5cm thick and 7.6cm
diameter. It is made of hard rubber and weighs about 170 grams. The same size
puck is used at every level. Sometimes a 250-gram puck is used for strength
training.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The same ball is used throughout the match unless it
becomes defective.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">When the puck leaves play or becomes defective, a new
one replaces it.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Players need shoes with little knives on the bottom
called cleats.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Players need shoes with one big knife on the bottom
called a skate.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 5; mso-yfti-lastrow: yes;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Protective equipment includes shin guards, a mouth guard,
and a groin cup.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Protective equipment includes shin guards, a groin
cup, a visored helmet, upper body pads, padded pants,<span style="mso-spacerun: yes;"> </span>padded gloves, and a mouth guard.<br />
<br />
All players have a fibreglass stick.<br />
<br />
Goalies have a larger stick, one special glove with a net, a helmet that
covers the entire head, thicker pads, and a neck guard.</p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed; width: 590px;">
<tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Soccer (Football)</span></b></p>
</td>
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Hockey (Ice Hockey)</span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Team Composition</span></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Team Composition</span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">There are 10 starting players, 1 goalkeeper and 1
backup goalkeeper. Between 3 and 12 substitutes are available for each team.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">There are 20 players on a hockey team, including 12 forwards,
6 defenders, 1 goalkeeper, and 1 backup goalkeeper.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><i>Only a fixed number of substitutions may be
made, typically 3 and never more than 5.</i></b></p>
<p class="TableContents"><b><i> </i></b></p>
<p class="TableContents"><b><i>This includes substitutions for injury and
substitutions between halves.</i></b></p>
<p class="TableContents"><b><i> </i></b></p>
<p class="TableContents"><b><i>Substitutions may only happen during stoppages.</i></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><i>Substitutions may be made any number of times,
even during play.<br />
<br />
This is done so frequently that no player other than the goalie is typically
on the ice for more than 60 seconds in a row.</i></b></p>
<p class="TableContents"><b><i> </i></b></p>
<p class="TableContents"><b><i>Groups of players called ‘lines’ are typically
on the ice together. For example, a player might not just be a center, they
would be a 3rd-line center, whom usually plays alongside two 3rd-line forwards.</i></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Typically 10 running players and 1 goalkeeper are
allowed on the field at once.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Typically 5 skating players and 1 goalkeeper are
allowed on the ice at once.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 5;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The goalie may elect to leave their assigned area at
any time and behave as if they were another non-goalie player.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">The team may choose to remove or ‘pull’ their goalie
at any time to allow an additional skating player.</p>
<p class="TableContents"> </p>
<p class="TableContents">This is common practice for the losing team near the
end of the game.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 6; mso-yfti-lastrow: yes;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Teams play in one several formations depending on the general
strategy they wish to employ.</p>
<p class="TableContents"> </p>
<p class="TableContents">A popular formation is “4-4-2 flat” , or 4 defenders,
4 midfielders, and 2 forwards, in that order.</p>
<p class="TableContents"> </p>
<p class="TableContents">A much more aggressive, complex formation is “3-4-3”,
which uses only 3 defenders, but 3 forwards.</p>
<p class="TableContents"> </p>
<p class="TableContents">Others include the 3-5-2, 4-3-3, and 4-5-1 formations.</p>
<p class="TableContents"> </p>
<p class="TableContents"> </p>
<p class="TableContents">Sources: <a href="https://www.soccermaniak.com/soccer-formations.html"><span color="windowtext" style="text-decoration: none; text-underline: none;">https://www.soccermaniak.com/soccer-formations.html</span></a></p>
<p class="TableContents"> </p>
<p class="TableContents"><a href="https://www.soccermaniak.com/3-4-3-soccer-formation.html"><span color="windowtext" style="text-decoration: none; text-underline: none;">https://www.soccermaniak.com/3-4-3-soccer-formation.html</span></a></p>
<p class="TableContents"> </p>
<p class="TableContents"> </p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Formations change by situation, given that the rink is
relatively small, especially with skates making everything faster.<br />
<br />
There are usually 3 forwards and 2 defenders on the ice at a time.</p>
<p class="TableContents"> </p>
<p class="TableContents">When trying to kill time during a penalty, a team
typically uses 2 forwards and 2 defenders. They will often form a square
‘box’ in the defensive zone to block shots and steal the puck.<br />
<br />
When the goalkeeper is removed, a team uses 4 forwards and 2 defenders. The
defenders will wait and the ends of the blue line to stop the puck and
prevent offsides.</p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7SLhOmTkZyq7ou9qJkj96CeaR6zgRvyuI68xnRdByXxy_u56b_ymW6qxi4khewtI8fTCc_2FqHRUGvU791TKv4Nwsv9U15z6lldfVOjglwRb6T62AHx_rKPo2sbCN3IfopLNtr3SuRTMK/s527/Pups+on+Ice.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="262" data-original-width="527" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7SLhOmTkZyq7ou9qJkj96CeaR6zgRvyuI68xnRdByXxy_u56b_ymW6qxi4khewtI8fTCc_2FqHRUGvU791TKv4Nwsv9U15z6lldfVOjglwRb6T62AHx_rKPo2sbCN3IfopLNtr3SuRTMK/s0/Pups+on+Ice.png" /></a></div><p></p>
<p class="MsoNormal"> <span> </span><span> </span><span> <span> </span><span> </span>Source: NHL. "Gritty's Pups on Ice"</span></p>
<p class="MsoNormal"> </p>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: medium none; mso-border-alt: solid windowtext .5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed; width: 590px;">
<tbody><tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Soccer (Football)</span></b></p>
</td>
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 18pt;">Hockey (Ice Hockey)</span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Penalties and Rules</span></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><span style="font-size: 16pt;">Penalties and Rules</span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Infractions are punished with yellow cards and red
cards. There is no immediate consequence of a yellow card, however a player
with a yellow card that commits an additional offense may recieve a red card.</p>
<p class="TableContents"> </p>
<p class="TableContents">A player with a red card is removed from play for the
rest of the match, and their team must continue the match with one fewer
player.</p>
<p class="TableContents"> </p>
<p class="TableContents">In some tournament formats, yellow cards persist
between matches.<b><span style="font-size: 16pt;"></span></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Infractions are punished with 2-minute minor
penalties, 4-minute double minor penalties, and 5-minute major penalties.</p>
<p class="TableContents"> </p>
<p class="TableContents">During a penalty, the punished player must leave the
ice, and the punished team must play with one fewer player on the ice. The
benefitting team is said to have a “power play”.</p>
<p class="TableContents"> </p>
<p class="TableContents">Minor penalties expire early when the team with a
power play scores a goal. Major penalties do not expire early.</p>
<p class="TableContents"> </p>
<p class="TableContents">Punishments are independent and do not accumulate
within a game or between games.<b><span style="font-size: 16pt;"></span></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><i>Fighting is not allowed and will probably result
in a red card and other disciplinary action.</i></b></p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><i>One-on-one fighting results in a 5-minute
penalty for both teams. Since both teams have a player in the penalty box,
both teams continue to play with five players and a goalie on the ice.</i></b></p>
<p class="TableContents"><b><i> </i></b></p>
<p class="TableContents"><b><i>Fighting must be a one-on-one affair, and can
only be done with bare fists. Sticks, skates, and other equipment may not be
used as weapons.</i></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Players may not intentionally touch the ball with any
part of their arms beyond the shoulder.</p>
<p class="TableContents"> </p>
<p class="TableContents">Goals and goal-scoring plays that involve a rebound
off a player’s arms are disallowed.</p>
<p class="TableContents"> </p>
<p class="TableContents">Handling the ball is not prohibited in cases where the
ball is rebounded off a player’s head or feet, or when the player is using
their hands to break a fall.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Players may touch the puck with any part of their
body, but certain actions are not allowed.</p>
<p class="TableContents"> </p>
<p class="TableContents">Players may not pass to teammates with their hands.
Players may pass with their skates, but such actions are rare.</p>
<p class="TableContents"> </p>
<p class="TableContents">Goals in which the puck was thrown or kicked into the
net are disallowed. Rebounds off skates<span style="mso-spacerun: yes;">
</span>that do not involve a kick are allowed.<br style="mso-special-character: line-break;" />
<br style="mso-special-character: line-break;" />
</p>
<p class="TableContents">Players may, however, use their hands to take a
mid-air puck and move it to their own stick.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 5;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">A player is in an offside position if they are nearer
the opponent’s goal line than all opposing players (except the goalkeeper).</p>
<p class="TableContents"> </p>
<p class="TableContents">If a player in an offside position is involved in a
play by their team, play is stopped.</p>
<p class="TableContents"> </p>
<p class="TableContents">Offsides do not apply to goal kicks, throw-ins, or
corner kicks.</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">A player is in an offside position if they are nearer
the opponent’s goal than both painted blue lines. (This area is called the
‘offensive zone’ to the player and the ‘defensive zone’ to the opponent)</p>
<p class="TableContents"> </p>
<p class="TableContents">If the puck enters the offensive zone when a player is
offside, play is stopped.</p>
</td>
</tr>
<tr style="mso-yfti-irow: 6;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><i>Physical contact is allowed provided that it is nopt
careless, reckless, or with excessive force according to the referee.</i></b></p>
<p class="TableContents"><b><i> </i></b></p>
<p class="TableContents"><b><i>Prohibited contact includes charging, jumping
at, tripping, tackling, striking, or pushing someone.</i></b></p>
<p class="TableContents"><b><i> </i></b></p>
<p class="TableContents"><b><i> </i></b></p>
<p class="TableContents"> </p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents"><b><i>Physical contact in the form of ‘checking’ or a
controlled tackle is allowed against a non-goalie player controlling the puck
(or was controlling it within the last half a second).</i></b></p>
<p class="TableContents"><b><i> </i></b></p>
<p class="TableContents"><b><i>The checking player may not use their stick or
skates to check, they may not check a player from behind or in an otherwise
vulnerable position, they may not aim for the head or the legs, and they may
not allow their feet to leave the ice during the check.<br />
<br />
Checking into the boards is allowed.<br />
<br />
Knocking an opponent over is relatively common, and isn’t prohibited.</i></b></p>
</td>
</tr>
<tr style="mso-yfti-irow: 7; mso-yfti-lastrow: yes;">
<td style="border: 1pt solid; mso-border-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Luis Suarez bites people. ( <a href="https://www.youtube.com/watch?v=Uy31pdfntUw"><span color="windowtext" style="text-decoration: none; text-underline: none;">https://www.youtube.com/watch?v=Uy31pdfntUw</span></a>
, at 0:35, 0:48, 1:17)</p>
</td>
<td style="border-color: currentcolor; border-style: none solid solid none; border-width: medium 1pt 1pt medium; mso-border-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-top-alt: solid windowtext .5pt; padding: 0cm 5.4pt; width: 221.1pt;" valign="top" width="295">
<p class="TableContents">Brad Marchand licks people. ( <a href="https://www.youtube.com/watch?v=Pgi3Gv9uqjA"><span color="windowtext" style="text-decoration: none; text-underline: none;">https://www.youtube.com/watch?v=Pgi3Gv9uqjA</span></a>
, at 1:04)</p>
</td>
</tr>
</tbody></table>
<p class="MsoNormal"> </p>
<p class="MsoNormal"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjehdPzLFuo26tfXsFlXjLCwEWEVOWQjqSIkuUyT2B_RLt845t1eR4wUofPBd947sSfywBoZXhpmz23dSVeFZh2Vz3nYMuZefVdRGX2sB0f0eiMIwRF92qBXGd2bHQL5BcCEFD22URkMbAc/s1024/puppy+cup+brasil.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="683" data-original-width="1024" height="342" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjehdPzLFuo26tfXsFlXjLCwEWEVOWQjqSIkuUyT2B_RLt845t1eR4wUofPBd947sSfywBoZXhpmz23dSVeFZh2Vz3nYMuZefVdRGX2sB0f0eiMIwRF92qBXGd2bHQL5BcCEFD22URkMbAc/w512-h342/puppy+cup+brasil.jpg" width="512" /></a></div><p></p>
<p class="Standard"><span> </span><span> </span><span> </span><span> </span><span> </span> <span class="photo_usage-attribution is-block" data-v-53d63e5d="" id="attribution"><a data-v-53d63e5d="" href="https://www.flickr.com/photos/73886864@N08/28682161837" rel="noopener" target="_blank">"DC Puppy World Cup"</a> <span data-v-53d63e5d="">
by
<a data-v-53d63e5d="" href="https://www.flickr.com/photos/73886864@N08" rel="noopener" target="_blank">Miki J.</a></span>
is licensed under
<a class="photo_license" data-v-53d63e5d="" href="https://creativecommons.org/licenses/by-nc-nd/2.0/?ref=ccsearch&atype=rich" rel="noopener" target="_blank">
CC BY-NC-ND 2.0
</a></span></p>
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<p class="TableContents"><b><span style="font-size: 18pt;">Soccer (Football)</span></b></p>
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<p class="TableContents"><b><span style="font-size: 18pt;">Hockey (Ice Hockey)</span></b></p>
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<p class="TableContents"><b><span style="font-size: 16pt;">Fan Experience</span></b></p>
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<p class="TableContents"><b><span style="font-size: 16pt;">Fan Experience</span></b></p>
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<p class="TableContents">On average, teams score about 1.2 goals each in a
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<p class="TableContents">On average, teams score about 3 goals each in a
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<p class="TableContents">There is one top-tier league called the EUFA
Champion’s League and several national professional leagues like the Premier
League (England), Serie A (Brazil), and La Liga (Spain).</p>
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<p class="TableContents">There are interleague matches, and EUFA is made of top
teams from their respective national leagues in Europe.</p>
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<p class="TableContents">There are no games between the leagues, but some
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ice. It gets a lot easier with time, but even players lose track of it
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<p class="TableContents">Don’t be intimidated and use the player movements as
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<![endif]-->Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-83097241597439042792020-07-27T17:39:00.003-07:002021-02-14T19:05:01.963-08:00The Price of Carbon Absolution<div>Worrying about <b>everything else</b> is wearing down my
sanity. Worrying about climate change feels like a welcome, familiar
distraction because the solutions are so linear. We can literally money our way
out of this one, but it's a lot of money? How much, exactly, is the personal price of carbon absolution?<br /></div><div><br /></div><span><a name='more'></a></span><div><br />
</div><p class="MsoNormal"><i>(Correction: The example stock used here, RNW, has some legacy projects in natural gas, which invalidate some of the calculations below, however those are being phased out and everything in their 5-year horizon is solar, wind, and geothermal.)</i></p><p class="MsoNormal"><i></i><br />Yes, innovations need to happen, and habits need to change, but it's a global problem and you are but one person. So, at an
individual level, for most of us, it comes down to throwing money at the
problem to make it go away.</p>
<h3 class="MsoNormal" style="text-align: left;"><b>What is the price of carbon absolution?<br /></b></h3>
<p class="MsoNormal">How much carbon is offset (if any), by owning $1 of a clean
energy company for 1 year? For example, Transalta Renewables (RNW on the Toronto
Stock Exchange) produced 1,173 GWh from renewable sources from 2020-01-01 to
2020-03-31 (source: <a href="https://www.transaltarenewables.com/2020/05/12/transalta-renewables-reports-first-quarter-2020-results/">https://www.transaltarenewables.com/2020/05/12/transalta-renewables-reports-first-quarter-2020-results/</a>
). Making the wrong, but useful assumption that these three months are representative
of the whole year gives us about 4,700 GWh, or 4.7 billion kWh for the year
2020. <span style="mso-spacerun: yes;"> </span></p>
<p class="MsoNormal">According to the US Energy Information Administration, US
electric generation releases about 450 grams of CO2 per kWh (source: <a href="https://www.eia.gov/tools/faqs/faq.php?id=74&t=11">https://www.eia.gov/tools/faqs/faq.php?id=74&t=11</a>
), based on 63% of the US's electricity production coming from fossil fuels.</p>
<p class="MsoNormal">Again making a wrong, but useful, assumption that the electricity
sources being replaced by a renewable source are 63% based of fossil fuels, and
that the renewable sources are themselves carbon neutral, we estimate that
every 1 kWh produced by a renewable keeps 450 grams of carbon in the ground.</p>
<p class="MsoNormal">That all means that a company like Transalta Renewables,
assuming it's not doing any creative accounting, is offsetting 2,115,000 metric
tons of carbon every year.</p>
<p class="MsoNormal">How much does it cost to attain these offsets? The market capitalization
(amount it would theoretically cost to buy the company) has been pretty stable
over the last couple years. Up to this point in 2020, it has averaged about $4 billion
CDN. </p>
<p class="MsoNormal">Cost to offset 1 ton / year = $4 billion market cap / 2,115,000
tons offset = $1,891. <br /></p>
<p class="MsoNormal">The average Canadian (and American, and Australian) emits 16
tons per year, as far as can be tracked. Therefore…</p>
<p class="MsoNormal"><b>If an average Canadian holds $30,260 of stock in a
company like RNW (or NPI, or INE), they will offset their carbon emissions.</b></p>
<p class="MsoNormal">It's a lot of money, but it's also not a sacrifice if you
can afford it; all of these stocks yield competitive dividends, even during the
(first?) year of the plague. Also, the market caps of these companies will
likely grow as they acquire or build more renewables, but their production will
likely increase proportionally.</p>
<p class="MsoNormal">Alternatively, one could purchase $160 USD (~$210 CDN) of
offsets every year from a place like b-e-f.org, at least while those relatively
cheap offsets exist. One could also purchase 16 tons worth of carbon permits on
the European Carbon Exchange (ECX), which are currently selling for 25 euros ($40
CDN) per ton each year, at an annual cost of $640 CDN/year. This also assumes
that the ECX is doing what it advertises to do, which isn't a perfect assumption.</p>
<p class="MsoNormal">In other terms, buying the stocks and holding them forever costs the same as 50 years worth of ECX permits, or 150 years of offsets at the cheapest available source. That's not including dividends, or the impact differences from removing all your carbon now vs removing it gradually.<br /></p>
<p class="MsoNormal">Speaking of assumptions, how robust is this $30,260 estimate
to violations of the assumptions used.</p>
<p class="MsoNormal">What if instead of 4,700 GWh of renewables per year, it's
4,000 – 5,000 GWh?</p>
<p class="MsoNormal">What if instead of a 63% fossil fuel mix being replaced,
it's a 40-80% mix?</p>
<p class="MsoNormal">What if instead of being carbon free, renewables emit 0-100
grams per kWh because of the production of raw materials like silicon, aluminum,
and especially concrete for hydroelectric dams?<br />
<br />
What if instead of emitting 16 tons, <span style="mso-spacerun: yes;"> </span>you
emit 10-20 tons?</p>
<p class="MsoNormal"><br />
At 4,000 GWh replacing a 40% fossil fuel source with a 100
grams / kWh source, that's an offset of 286 g /kWh from the fossil fuels, and 60 g/kWh from the displaced clean sources for a net offset of 246 g/kWh, meaning only 984,000 tons are offset, at a market cap cost of
$4,065 per ton per year.</p>
<p class="MsoNormal">At 5,000 GWh replacing an 80% fossil fuel (571 grams / kWh) source
with a 0 grams / kWh source, as much as 2,855,000 tons are offset, at a market
cap cost of $1,401 per ton per year.</p>
<p class="MsoNormal"><b>From best-case-all to worst-case-all, it would take $14,010
to $81,300 </b>of stock holdings not to undo your personal damage to the air, but
just to stop doing more.</p>
<p class="MsoNormal">That's a huge price, and it's a large part of why experts say
to 'reduce what you can, and offset the rest', rather than just trying to buy
absolution without changing your life. It is, however, a sane number. It's the
size of a down payment on a house, and like one it's a responsible step towards
a secure personal future.</p><p class="MsoNormal">Finally, use <a href="https://www.ecosia.org/">Ecosia</a> instead of Google, and avoid beef and pork, even just to prevent colon cancer.<br />
</p><p class="MsoNormal"><b><br /></b></p><p class="MsoNormal"><br /></p><h3 class="MsoNormal" style="text-align: left;"><b>Blog Update: You are already part of the solution.</b></h3>
<p class="MsoNormal">We had this discussion before in 2019, here: <a href="https://www.stats-et-al.com/2019/10/offsetting-carbon-emissions-of-blog-and.html">https://www.stats-et-al.com/2019/10/offsetting-carbon-emissions-of-blog-and.html</a>
. That post has a literature search to determine how much CO2 is emitted from
hosting a web page. The estimate I landed on was about 1 gram per page view.</p>
<p class="MsoNormal">Also in 2019, I purchased 3 tons to offsets from the Bonneville
Environmental Foundation in order to offset 50 grams per view of the blog's
first 60,000 views. It cost $30 to do so, and that was offset in part from the
ads that were being delivered to the most lucrative 10% of viewers.</p>
<p class="MsoNormal">In 2020, Google took away the ability to deliver ads to only
some of my readers, so I took away Google's ability to deliver ads to any of my
readers.</p>
<p class="MsoNormal">I also purchased offsets for 50 grams per view for views 60,001
to 100,000, again from <a href="http://b-e-f.org">b-e-f.org</a> , as shown below. <br /></p>
<p class="MsoNormal"> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVWtru-WbXTYPIel053JGQ2lbE6e39B0c1cw_1ctmiO3slx9kT-2uaekNsrHYHuDtTbMoCA5ta8aaSPIqsI_N0M7MgcjoOrRRJ00kaUT-yNyV2gQHa5ykwps42_mu9X0_XNfg1uj649oLW/s1383/Carbon+Offset+Tons+4+5.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="549" data-original-width="1383" height="199" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVWtru-WbXTYPIel053JGQ2lbE6e39B0c1cw_1ctmiO3slx9kT-2uaekNsrHYHuDtTbMoCA5ta8aaSPIqsI_N0M7MgcjoOrRRJ00kaUT-yNyV2gQHa5ykwps42_mu9X0_XNfg1uj649oLW/w500-h199/Carbon+Offset+Tons+4+5.PNG" width="500" /></a></div><br /><p class="MsoNormal">
This will be <strike>100</strike> 150 grams per view going forward. The certificates for the<strike> 5 </strike> 15
additional tons is here:</p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGjRv38tUdffXcjQZ8KE4Y6BuIM8ss0Kxzco9kcs8rx2K4uweBmzIew5y7ayZvzYhvlVyOcIEBFx4zTcIHKgkYRj9BkIUKYIqW6ozPDDB5PvDl9RWpuuQ88UY8sU1XR9sKsPLbTFB5QC7r/s1321/BEF+Carbon+Tons+6-10.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="511" data-original-width="1321" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGjRv38tUdffXcjQZ8KE4Y6BuIM8ss0Kxzco9kcs8rx2K4uweBmzIew5y7ayZvzYhvlVyOcIEBFx4zTcIHKgkYRj9BkIUKYIqW6ozPDDB5PvDl9RWpuuQ88UY8sU1XR9sKsPLbTFB5QC7r/s640/BEF+Carbon+Tons+6-10.PNG" width="640" /></a></div><p class="MsoNormal"><br /> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOnZ-_lU8l65zjlfIZN5Y_Ii_pDKyniVxPeiembgRCgXjhUTABx5713BldqLg0ent2Ueg_QyadjIEntmwoamHXrzSW7-apnitqVMEkup8dOgCGU7q7gMvdzsZzLlHPAfoBEhXuhYDIHiHG/s1501/Stats+Et+Al+Carbon+Offsets+Tons+11-20.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="569" data-original-width="1501" height="242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOnZ-_lU8l65zjlfIZN5Y_Ii_pDKyniVxPeiembgRCgXjhUTABx5713BldqLg0ent2Ueg_QyadjIEntmwoamHXrzSW7-apnitqVMEkup8dOgCGU7q7gMvdzsZzLlHPAfoBEhXuhYDIHiHG/w640-h242/Stats+Et+Al+Carbon+Offsets+Tons+11-20.png" width="640" /></a></div><br /><p></p><p class="MsoNormal">Here is the David Suzuki Foundation's <a href="https://davidsuzuki.org/wp-content/uploads/2019/10/purchasing-carbon-offsets-guide-for-canadians.pdf">review of Canadian carbon offset sellers</a>. An ideal Canadian carbon sink is probably coming in the form of a seaweed farm by <a href="https://www.cascadiaseaweed.com/">Cascadia Seaweed.</a> but we don't have time to be picky, so American offsets it is. <br /></p><p class="MsoNormal"><br /><br /></p><p class="MsoNormal"><b>Footnote on ads</b></p>
<p class="MsoNormal"><br />
Removing ads wasn't done out of altruism, it was a self-serving move. It didn't
make sense to write all this to build a professional image for myself if it was
going to be interrupted every paragraph by SAVE THE PRINCESS, TEST YOUR IQ WITH
THIS GAME FOR YOUR MOBILE DEVICE. </p>
<p class="MsoNormal"> </p>
<p class="MsoNormal"><b>Footnotes on company choice</b></p>
<p class="MsoNormal">This company doesn't do much besides produce renewable energy;
it doesn't manufacture renewable infrastructure, it just buys it and sells the electricity,
so it's nearly as close to buying solar/wind/hydro and paying someone to
maintain it as you can get in the stock market.</p>
<p class="MsoNormal">Transalta Renewables (RNW) has the complication of its
relationship with Transalta, which produces power through a variety of means,
including dirty ones, but RNW itself is pretty clean.</p>
<p class="MsoNormal">Northland Power (NPI) and Innergex Renewable Energy (INE) have
similar business models, but without the parent company complication. I just
found the data on RNW to be easier to parse and check.</p>
<p class="MsoNormal">Conflict of interest statement: I hold a long position in
all of these stocks, although not one large enough to be carbon neutral without
additional measures.</p>
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<![endif]-->Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-84854930143687704332020-07-05T15:40:00.005-07:002021-01-30T14:37:06.293-08:00Statistics, Gambling, and Games of ChanceThis is a proposal for a survey course on statistics that uses
gambling extensively in examples. The target audience is senior undergraduates with a non-statistical
background, but quantitative students will also find enough novelty to be
interested. The goal of the course is not to encourage gambling, but to use it
as a vehicle for a broad range of otherwise difficult statistical topics.<br />
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="mso-spacerun: yes;"></span></div>
<a name='more'></a><br />
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<br /></div>
<div class="MsoNormal">
For example, roulette is used to deliver a lesson on
expected value, but many of the bets you can make in roulette have the same
expected value despite the widely different payouts. Likewise, the probability
of poker hands can be calculated exactly with combinatorics, and the odds offered
on sport events can be used to explain log-odds, the basics logistic regression,
and confidence intervals.</div>
<div class="MsoNormal">
There is opportunity for some larger projects, like
predicting sports, but a lot of the smaller assignment questions could be
're-skinned' from probability courses.</div>
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<br /></div>
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Because of the wide range in depth, and the diversity in prospective
students, as well as the stigma of gambling and of teaching people to gamble, this
would probably be best as a zero-credit course or an independent lecture
series. However, it could also be used as an outreach tool to bring talent into
statistics and data science that might not otherwise consider it.</div>
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<br /></div>
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The syllabus is given in parts because the timing hasn't
been worked out yet. This could be anywhere from 20 to 80 hours of lectures
depending on pacing and topic selection.</div>
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<br /></div>
<h4 class="MsoNormal">
<span style="font-size: large;">Draft Syllabus:</span></h4>
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<br /></div>
<div class="MsoNormal">
<b>Part 1: Preliminaries</b></div>
<div class="MsoNormal">
- What
is gambling?</div>
<div class="MsoNormal">
- PDF,
CDF, continuous and discrete distributions</div>
<div class="MsoNormal">
- Means,
medians, variance, and skew</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Part 2: Classic games of chance</b> <br /></div><div class="MsoNormal">- Monty
Hall problem: Sample spaces</div>
<div class="MsoNormal">
- Random
packs of cards, loot boxes: Geometric and negative binomial distributions</div><p>
- Craps, Yahtzee, dice games: Independent
events, binomial distribution, normal approximation </p><p></p><p>- <a href="https://www.stats-et-al.com/2020/12/borel-dice-edition-brute-forcing.html">Simulating dice events with Borel</a>.<br />- <a href="https://www.stats-et-al.com/2019/04/bingo-analysis-tutorial-in-r.html">Bingo and discrete distributions.</a><br /></p><p>- Roulette, expected value, and
house advantage (See also Vig/Juice)
<br />
</p><div class="MsoNormal">
- Blackjack
and Transition Matrices (See also <a href="https://www.stats-et-al.com/2017/11/snakes-and-ladders-and-transition.html">Snakes and Ladders</a>, Monopoly)</div><div class="MsoNormal">- <a href="https://www.stats-et-al.com/2021/01/lottery-tickets-baseball-cards-and.html">The Coupon Collector's Problem, trading cards, and Magic cards</a> <br /></div>
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<br /></div>
<div class="MsoNormal" style="tab-stops: 168.75pt;">
<b>Part 3: Combinatorics<span style="mso-tab-count: 1;"> </span></b></div>
<div class="MsoNormal">
-
Counting principles, factorials, and Stirling's approximation.</div>
<div class="MsoNormal">
- Combinations
and Permutations.</div>
<div class="MsoNormal">
- Playing
Cards: conditional probability, information, and 'blockers'.</div>
<div class="MsoNormal">
- Blackjack
again: Counting cards, shoes, and asymptotics.</div>
- The hypergeometric distribution.<span style="mso-tab-count: 1;"> </span><br />
<div class="MsoNormal">
- Poker:
Straights, flushes, pairs, and triples.<span style="mso-tab-count: 1;"> </span></div>
<div class="MsoNormal">
<span style="mso-tab-count: 1;"></span>- Poker:
Accounting for community cards, and for drawing rounds.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Part 4: Utility</b></div>
<div class="MsoNormal">
-
Marginal utility, utility curves.</div>
<div class="MsoNormal">
-
Lotteries and Insurance: (different utility for you and the company).</div>
<div class="MsoNormal">
-
Future value, present value, discounting rates.</div>
<div class="MsoNormal">
<br /></div>
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<b>Part 5: Prediction Markets and Sports Betting</b></div>
- Decimal vs American Odds: log-odds,
logistic regression, ordinal logistic regression
<br />
- Odds
movements, stock price movements: time-series models<br />
-
Parlays and correlated bets<br />
- Over/Unders, Vig/Juice, Exotic
or Prop bets<br />
- Pre-game and in-game betting: Probabilities
changing over time<br />
- <a href="https://www.stats-et-al.com/2021/01/fantasy-sports-explained.html">Fantasy sports</a>: expected and
variance of sums, opportunity cost<br />
- Hedging strategies and arbitrage<br />
<div class="MsoNormal" style="text-indent: 36pt;">
<br /></div>
<div class="MsoNormal">
<b>Part 6: Common sports models</b></div>
<div class="MsoNormal">
- Soccer
and Hockey: Poisson, Zero-Inflated Poisson, and Diagonally-Inflated Poisson</div>
<div class="MsoNormal">
-
Basketball: Bivariate normal</div>
<div class="MsoNormal">
- Head-to-head
matchups (team sports): Elo, Glicko, and other ratings systems</div>
<div class="MsoNormal">
-
Many-player matchups (races): Microsoft TrueSkill</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Part 7: Game theory for player-vs-player</b></div>
<div class="MsoNormal">
- Zero-Sum
games vs win/win (Nash equilibria)</div>
<div class="MsoNormal">
- Rock-paper-scissors</div>
<div class="MsoNormal">
-
Prisoner's dilemma</div>
- Jeopardy: Strategy and Decision
flowcharts<br />
<br />
<div class="MsoNormal">
<b>Part 8: Texas Hold'em<span style="mso-tab-count: 1;"> </span></b></div>
- Pot odds, implied odds, the fundamental
theorem of poker<br />
- Game Theory Optimal, Ranges,
Bluffing<br />
- Independent Chip Model<br />
<div class="MsoNormal" style="text-indent: 36pt;">
<br /></div>
<div class="MsoNormal">
<b>Part 9: Bankroll management</b></div>
<div class="MsoNormal">
- Kelly
Stakes Criterion, Bayesian adjustments, Simultaneous Kelly</div>
<div class="MsoNormal">
-
Drawdown, Value-At-Risk, Extreme Value Functions</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Part 10: Simulations and Machine Learning</b></div>
- Monte Carlo simulations, Monte Carlo
Markov Chain
<br />
<div class="MsoNormal">
- How
AIs for games like backgammon are trained</div>
<div class="MsoNormal">
-
Minimax algorithms</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Historical / Cultural hooks</b></div>
- The birth of probability comes
from an unresolved game of chance
<br />
<div class="MsoNormal">
- Chaturanga,
a very old version of chess which used dice to determine pieces to move.</div>
- Parcheesi is an older, dice-based
version of Sorry!<br />
- The MIT Blackjack team<br />
- The Royal Game of Ur<br />
- Hubbub and Chekutnak: Indigenous
Games of Chance<br />
- Mahjong and trick-taking games.<br />
<div class="MsoNormal">
<br /></div>
<h4 class="MsoNormal">
<span style="font-size: large;"><span style="line-height: 107%;">Draft Introduction:</span></span></h4>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>What is Gambling</b></div>
<div class="MsoNormal">
What legally constitutes gambling changes with locations and
time. By some definitions, anything that involves a degree of skill (e.g.
choosing players in a fantasy sports league) is a competition and therefore not
gambling. For the purposes of this class, we'll use the opposite extreme: Everything
with financial uncertainty is gambling, including things aren't commonly
considered to be games at all.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Buying a home is the stereotypical 'responsible purchase',
and yet it's an *enormous* gamble. The long-term value of a house has a lot of
variance. If there is variance, there is risk, and for the purposes of this
course this gambling.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Most investments carry financial risk, but unlike most games
of chance they carry relatively low risk and a positive expected value. Even
bonds, which are just loans to corporations or governments, even if re-payment is
guaranteed, carry risk in their resale value related to exchange rates.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Actuarial science is a whole field, closely related to
statistics, that deals with financial uncertainty. Traditionally it deals with
insurance and the premium payments that support it, but it also deals with the
investment of funds until they are needed for payments of claims. If you enjoy
this course and wish to pursue further theoretical depth, it's recommended that
you look further into statistics and especially actuarial science.</div>
<div class="MsoNormal">
<br /></div>
<span style="font-size: large;">
</span><br />
<h4>
<span style="font-size: large;">Reading Sources:</span></h4>
<div class="MsoNormal">
<b>
<br style="mso-special-character: line-break;" />
</b></div>
<div class="MsoNormal"><a href="https://www.stats-et-al.com/2020/09/review-of-theory-of-gambling-and.html">"The Theory of Gambling and Statistical Logic" by Richard Epstein. <br /></a></div><div class="MsoNormal"> "Weighing the Odds in Sports Betting" by King Yao</div>
<div class="MsoNormal">
<a href="https://www.stats-et-al.com/2019/08/reading-questions-struck-by-lightning.html">"Struck by Lightning" by Jeff Rosenthal</a></div>
<div class="MsoNormal">
"The Theory of Poker" by David Sklansky <br /></div>
<div class="MsoNormal">
Indigenous Games: <a href="http://mathcentral.uregina.ca/RR/database/RR.09.00/treptau1/">http://mathcentral.uregina.ca/RR/database/RR.09.00/treptau1/</a></div>
<div class="MsoNormal">The Game of Borel.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
If this course proposal gives you ideas as an educator, see
also my previous course proposal on "<a href="https://www.stats-et-al.com/2018/11/statistics-in-politics-and-demographics.html">Statistics in Politics and Polling</a>"
<span style="mso-spacerun: yes;"> </span>for a similar outreach / cross-disciplinary
approach.</div>
<div class="MsoNormal">
<a href="https://www.stats-et-al.com/2018/11/statistics-in-politics-and-demographics.html">https://www.stats-et-al.com/2018/11/statistics-in-politics-and-demographics.html</a><br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
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<![endif]-->Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-68186548569920294472020-06-30T12:56:00.000-07:002020-06-30T19:23:58.372-07:00Cheating vs Innovation in Sports<br />
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Why do some changes in <span>sports</span> end up being considered <span>cheating</span>, and others <span>innovation</span>? Let's look at some historical examples for patterns.</div>
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In baseball, "the shift" is a strategy in which
defensive players deviate, or ‘shift’ from the default locations for their
positions to locations closer to where they expect the ball to land. This
practice has had a measurable statistical effect on the game; hits other than
home runs have become rarer, and the hits that do happen are more often singles
compared to seasons before 2010. The shift is simply data-driven strategy and
yet the practice is still controversial.</div>
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It seems like such an arbitrary thing to call out as unfair </div>
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<a name='more'></a><br />
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<h4 class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;">The
Defensive Shift</span></h4>
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Let's start with the shift. In the 2010, almost nobody used
it. By 2019, almost everybody did, and players carried index cards of where
they should be in anticipation of each batter. <span style="mso-spacerun: yes;"> </span>In 2015, the league commissioner publicly
talked about banning the shift ( <a href="https://www.si.com/mlb/2015/01/26/rob-manfred-defensive-shifts-mlb-commissioner">https://www.si.com/mlb/2015/01/26/rob-manfred-defensive-shifts-mlb-commissioner</a>
) and the mechanics of the ban were still being discussed in 2018 ( <a href="https://www.si.com/mlb/2018/07/25/defensive-shifts-official-baseball-rules">https://www.si.com/mlb/2018/07/25/defensive-shifts-official-baseball-rules</a>
<span style="mso-spacerun: yes;"> </span>)</div>
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The movement of fielders isn’t new. The fielding players
have always had the freedom to move about into each others’ traditional areas,
which is why calling for the ball in pop-up is a fundamental skill taught at
the little-league level for kids 12 and younger. Those traditional placements
came about so that there would always be someone near where the ball would be
hit, but those placements were decided upon before we collectively had data on
the tendencies of every individual hitter.</div>
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It’s unclear how a ban on the shift would be enforced. Would
they paint bright pink spots where the outfielders would have to stand until
the ball was hit? Would there be marked-off territory for each fielder to be
confined within? Would they ban the notes cards that tell fielders where to be
before each batter appears? Would there need to be additional umpires in the
outfield? (MLB baseball already uses three umpires and video review).</div>
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The shift is staying; it’s a part of the game at least until
batters learn to either be less predictable, or to hit a home run every time.
Why was there any controversy at all? My guess is the newness of the practice
(circa 2015) compared to the oldness of modern baseball (circa 1860). As the
reaction to the shift shows us, age plays a big part in the reason some changes
in play become ‘innovations’ and others become ‘cheating’.</div>
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<h4 class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;">Before the concrete sets: Ball Dimples
and Belly Putters</span></h4>
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There seems to be balance between how new a sport is how big
of a change in being made to the game. In the original Scottish version of
golf, golf balls were smooth and could rarely be hit further than 40 metres
because of air resistance. Today, amateurs hit golf balls more than 200 metres.
The secret is dimples; they minimize air resistance and maintain laminar flow
so that the ball flies in a smooth parabolic arc.</div>
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That one equipment change radically alters golf. If the ball
can be hit 5 as far, the courses need to be 5 times as long to maintain
difficulty. That’s 5 times as much land and 5 times as much maintenance. Any
pre-dimple course is laughably obsolete. Players with dimpled balls are playing
a different sport compared to their smooth-ball competitors. But dimples were
introduced to golf early enough that there wasn’t enough traditionalist
pushback to enforce smooth balls. The dimple stuck.</div>
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Fast-forward to 2011 and the adoption of the ‘belly putter’
by pros. A putter is the only kind of golf club allowed to be used in the
short-range portion of each hole, the green. Here, accuracy and
steadiness-of-hand are critical. There are very strict rules about the shape
and weight of the ‘face’, the part of the club that actually touches the ball,
but not so much the handle. The belly putter is just an ordinary putter with a
longer handle, which allows a player to press the end of the putter against
their gut and provide extra steadiness. By 2013 the major governing bodies of
golf had unanimously adopted a rule that forbade using your gut as an anchor. (
<a href="https://www.thestar.com/sports/golf/2013/05/21/belly_putter_controversy_golf_adopts_rule_to_outlaw_putters_anchored_to_the_body.html">https://www.thestar.com/sports/golf/2013/05/21/belly_putter_controversy_golf_adopts_rule_to_outlaw_putters_anchored_to_the_body.html</a>)</div>
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I wonder what the anti-dimple crowd would have said about
that ruling.</div>
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<h4 class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;">Pace of Play: Dribbling and The Trapezoid</span> </h4>
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Sometimes things are considered cheating to preserve the
entertainment value of the sport. Belly-putting looks stupid and it’s less
impressive than regular putting. Famed NHL goalie Martin Brodeur regularly used
a tactic in which he would leave the net and take control of the puck behind
the net in order to pass to a teammate. He was relatively safe from opposing
interference because of the protections afforded to goalies in hockey and
soccer. This tactic was made illegal by the introduction of ‘the trapezoid’, a
marked region behind the net where the goalie cannot touch the puck. What
Brodeur was doing wasn’t dangerous, but it did make the game slower and less
exciting.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Basketball was designed to be a stand-and-pass game. It
underwent a nearly dimple-sized transformation with the addition of dribbling
or bouncing the ball on the ground. Dribbling was originally a hack; it was a
workaround by which you technically passed the ball to yourself. Dribbling
changes the game radically, but it was probably allowed because the sport was
very young at the time. Dribbling makes basketball a much faster, rougher, and
entertain sport than James Naismith originally designed, but at least it didn’t
invalidate all the basketball courts made before then.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<h4 class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;">Safety: The Fosbury Flop and Skating Backflips</span></h4>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Maybe dribbling was easier to accept because it was a change
in technique and not equipment. The high jump has been a competitive endeavor
since the discovery of legs by famed 12th century merchant Thomas Jump (modern
scholars actually believe he had merely copied the idea of jumping from the
Chinese). Until 1968, high jump competitors hopped over poles as if they were
hurdles: leaving the ground with their feet, staying upright, and then landing
on their feet. Then along came Dick Fosbury, who instead jumped at the bar
sideways and twisted his body around and above the pole like an inverted limbo
dancer, only to fall on the ground on his side. This became know and the
Fosbury Flop and it was mind-blowing. In the traditional jumping strategy, you
had to get your whole body over the bar at once. With the Flop, only a part of
your body needed to be above the bar at any time. A proper Flopper might clear
the bar without ever having their center of mass above the bar.</div>
<div class="MsoNormal">
</div>
<div class="MsoNormal">
The Fosbury Flop was so effective that other competitors saw
it and started practicing it while waiting for their turn to compete during the
event. ( <a href="http://archive.nytimes.com/www.nytimes.com/packages/html/sports/year_in_sports/10.20.html">http://archive.nytimes.com/www.nytimes.com/packages/html/sports/year_in_sports/10.20.html</a> )</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
After Fosbury painfully earned the high jumping gold medal
at the 1968 Mexico City Olympics, officials went along it and placed a soft
cushion next to the high-jump bar to mitigate the risk that comes with landing
scorpion-style on your face.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Often cheating just comes down to safety. The backflip is
banned from figure skating, even though it’s a lot like a Fosbury Flop. Maybe
ice skates would cut up the landing cushion.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Safety also explains why most performance enhancing drugs
are banned in most competitions. The ban reduces the incentive to use drugs
that could have problematic long-term effects. You might not consider chess a
sport, but the World Anti-Doping Agency considers it enough of one to ban
certain substances:<span style="mso-spacerun: yes;"> </span>Adderal, Ritalin,
Modafinil, and excessive levels of caffeine. </div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
( <a href="https://www.fide.com/FIDE/handbook/WADA%20Anti%20Doping.pdf">https://www.fide.com/FIDE/handbook/WADA%20Anti%20Doping.pdf</a>
<span style="mso-spacerun: yes;"> </span>)</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
In soccer, it used to be common to rush in and assault and
otherwise interfere with the goalie in order to score or help a teammate score.
This resulted in a lot of goalie injuries and protections for players have been
put in since. A lot of rule changes are meant for player safety, often taking
the choice away from players and coaches in order to eliminate the effect of
peer pressure on taking additional risks (e.g. in the case of mandatory
protective gear like helmets and cups). </div>
<div class="MsoNormal">
<br /></div>
<h4 class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;"> </span></h4>
<h4 class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;">Using Technology: Fancy
Swimsuits and the Houston Asterisks</span></h4>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Equipment requirements need to be updated regularly to
maintain safety and entertainment value. Acceptable sizes of hockey goalie pads
seem to be changed every few years. Equipment requires need to be updated even
when the dynamics of a sport don’t change. </div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
In the 2008 Beijing Olympic Games, swimmers wearing the
Speedo LZR Racer, a full-body swimsuit that “it compressed a swimmer’s body
into a streamlined tube and trapped air, adding buoyancy and reducing drag.
Speedo says 98 percent of the medals at the 2008 Olympics were won by swimmers
wearing the LZR.“ By 2009, full-body swimsuits were banned from competitions,
along with some materials. Competitive swimwear that covers an approved area
still allowed, and swimwear design is still actively being advanced, but new
designs need to be approved the governing swimming body, FINA, before being
allowed in races. ( <a href="https://web.archive.org/web/20150426215139/http:/www.fina.org/H2O/docs/rules/FRSA.pdf">https://web.archive.org/web/20150426215139/http://www.fina.org/H2O/docs/rules/FRSA.pdf</a>
<span style="mso-spacerun: yes;"> </span>) ( <a href="https://www.scientificamerican.com/article/how-speedo-created-swimsuit/">https://www.scientificamerican.com/article/how-speedo-created-swimsuit/</a>
)</div>
<div class="MsoNormal">
<br /></div>
The Houston Astros sign stealing scandal was a source of
armchair rage-judgement from the end of the 2019 season all the way up until
World War Flu came to America. In baseball, the pitcher and catcher work in a
partnership against the other team’s batter. The catcher squats close behind the
batter and uses hand signals to tell the pitcher how to throw the ball so that
they can, well, catch it. Since the batter needs to watch the pitcher at all
times, and since the catcher is behind the batter, the catcher usually has free
reign to communicate with the pitcher in secret.
<br />
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
You know what this needs a picture. I usually explain this
part with charades or pantomime, and *wiggles fingers with neon-painted nails*
doesn’t translate to text like you’d think.</div>
<div class="MsoNormal">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2cDF0VtauJVe8C8XOxpITlYdSUPZSjTyquHm7G9nUrPaPXmN2IiKWY_PNU8qLUOHIaRlteIZv67pBZNt7rUSaJ-VIJNN6SIhlF5BB2tR6CZ23HIQFhexM9KCul9z3R7MtDTaH4FD4jgKT/s1600/14069782514_3bd3a1f454_b.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="683" data-original-width="1024" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2cDF0VtauJVe8C8XOxpITlYdSUPZSjTyquHm7G9nUrPaPXmN2IiKWY_PNU8qLUOHIaRlteIZv67pBZNt7rUSaJ-VIJNN6SIhlF5BB2tR6CZ23HIQFhexM9KCul9z3R7MtDTaH4FD4jgKT/s400/14069782514_3bd3a1f454_b.jpg" width="400" /></a></div>
<br />
<br />
<span class="photousage-attribution"><a href="https://www.flickr.com/photos/48227962@N04/14069782514" target="_blank">"Swing
and a Miss"</a> <span data-v-106f2c17="">by <a data-v-106f2c17="" href="https://www.flickr.com/photos/48227962@N04" target="_blank">PMillera4</a></span> is licensed under <a data-v-106f2c17="" href="https://creativecommons.org/licenses/by-nc-nd/2.0/?ref=ccsearch&atype=rich" target="_blank">CC BY-NC-ND 2.0 </a></span><span style="mso-spacerun: yes;"> </span>from left to right, that's the Umpire (e.g. Judge,
Referee), Catcher, and Batter. The pitcher is off-camera where all three are
looking. See how the batter can't watch the catcher and batter at the same
time?
<br />
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
However, baseball is played with more than three people,
sometimes. Sometimes that batter has some help, like a runner on second base
who is in the perfect position to see *wiggles fingers* from the catcher. That
runner might notice that one type of wiggle signals a fastball to be thrown,
and that another wiggle signals a curveball to be thrown. The runner then tells
their batting teammate what throw to expect, thus conferring an advantage to
them. That reading and communicating of catcher’s signals is called
“sign-stealing”. It has been in baseball ever since catchers used signals.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
So what did the Astros do that was so bad? They stole signs
too well. Rather than hope for a runner on second, someone in the stands had a
very expensive camera trained on the opposing catcher’s hands to watch to
*wiggles fingers*. That hot info was sent to Astros’ teammates or employees who
would warn the batter about certain pitches with a loud whistle or the bang of
a trash can.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
This continued through the 2017, 2018, and 2019 seasons, as
found by regular old fans combing through broadcast footage after suspicions
got too high. Suspensions and millions of dollars in fines were handed out the
Astros, and the team’s ability to bring in top new players was reduced for a couple
of future seasons. Others called for the Astros to be stripped of their 2017
World Series win. Months of bedlam ensued.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
But what exactly was the cheat? If sign-stealing has been in
the game forever, what was the transgression? This time it wasn’t just a better
use of an old technique, but the introduction of new technology and an unequal
access to the advantage. A lot of the parts in this form of sign-stealing could
only be done in at home games in Houston. It introduced new equipment to the
game without the league’s consent. It made people sad.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
(For way more discussion on recent cheating, see 538's Hot
Takedown podcast episode on scandals of the century, including the Houston
Astros controversy)</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Is it the deception factor? That the Astros hid that they
had this privileged information, or that they were preventing the opposing
pitchers from deceiving Astros batters? Baseball has a complicated history with
deception.</div>
<h4>
<span style="font-weight: normal;"><span style="font-size: large;"><b> </b></span></span></h4>
<h4>
<span style="font-weight: normal;"><span style="font-size: large;"><b>Not asking permission: Stealing Bases and Pulling Goalies</b></span></span></h4>
<br />
<br />
<div class="MsoNormal">
Stealing bases (running from one base to the next when the
ball is still in pitcher’s hand) is okay. It’s only because somebody noticed
that it wasn’t explicitly banned, tried it in an early MLB game, and argued
that it should be allowed during the game. The umpires just went with it and
the rest is history.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
On the other hand, balking, a deceptive pitching strategy
meant as a disincentive to stealing* (and to disorient batters) results in a
one-base penalty.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
(For more on Stealing Bases, see 1st by SB Nation on Youtube
at <a href="https://www.youtube.com/watch?v=VdMPxmZ8CDU">https://www.youtube.com/watch?v=VdMPxmZ8CDU</a>
)</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Around the same time as the first stolen base, Art Ross
executed the brilliant idea in an NHL hockey playoff game in 1931. Near the end
of game that his Montreal Canadiens were losing by one goal to the Boston
Bruins, he decided to ‘pull the goalie’. When a coach pulls the goalie, the net
minder leaves play in exchange for an additional attacker. </div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
The team using this
tactic is still technically only keeping six players on the ice at a time, but
the goalie is governed by different rules and has different equipment. But what
if Art Ross had never tried exchanging a goalie for another skating, mobile
player? The first season of the NHL was the winter of 1917-18, so this was
season 14. Would goalie-pulling be banned if someone had first tried it in 1950
instead? 1980? 2020? (See: <i>Dan Diamond<cite><span style="font-family: "calibri" , sans-serif; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-hansi-theme-font: minor-latin;"> (2002). Total Hockey: The
Official Encyclopedia of the National Hockey League, Second Edition. New York:
total Sports Publishing. p. 204.)</span></cite></i></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<h4 class="MsoNormal">
<span style="font-size: 16.0pt; line-height: 107%;"><span style="font-size: large;">Tricking the
officials: Pitch Framing and Diving</span></span></h4>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Pitch framing, which is the intentional deception of the
officials in baseball is considered okay. It’s hard to enforce like the shift,
but part of the problem is that whether a pitch counts as a strike or not is
still a matter of human judgment.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Judgment rules are often thorny, and while often necessary,
they also often make the sport in question worse. Diving and faking injuries
and impacts, which is the intentional deception of officials to draw penalties
in soccer, results in a fine if you get caught. Diving sucks, but it is a winning
strategy because it exploits judgment. On the other side, when judgment is
removed, like the in the case of video review of goals, the entertainment
quality can suffer from delays. There’s really no winning if you’re a soccer
referee.</div>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Mention"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Smart Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hashtag"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<![endif]-->Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-69791166093262563722020-06-26T10:07:00.000-07:002020-06-26T10:07:00.185-07:00When to use "the" or "a" in scientific writing<br />
<br />
<br />
<br />
A.K.A. : "The",
the definitive definite article article.
<br />
<br />
<div class="MsoNormal">
<br /></div>
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"The", while making up about 7% of all written and
spoken English words, is the hardest word to get right. The rules
surrounding "the" are so difficult to define that comprehensive
dictionaries can spend 5 of more pages trying...</div>
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<br /></div>
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<br /></div>
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</div>
<a name='more'></a><br /><br />
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<br /></div>
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<i>This article is meant to be part of the preliminary reading in a
'Writing for Statistics and Data Science' course or 'Scientific Writing'
course. Either missing 'the' or 'a', or putting it in the wrong place
is the most common error I've seen when grading written assignments or
even when copyediting scientific journal articles.</i></div>
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<br /></div>
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<i>As
a teacher, it's a tough call whether to put this as an optional 'week
0' reading, or as part of a week 1 lecture. Optional work rarely gets
done (although in week 0-1 maybe things are quiet enough), but as a core
part of the course it's a pretty bland first impression.</i></div>
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<i><br /></i></div>
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<i>Article starts after the sprawling puppy.</i></div>
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<i><br /></i></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6e2LRxiYijblCyY3dGMQd23WmUXJtZHkiH3EDRZT55BomSfA5hLoU7QZLBg1nfwVIgLP5y4s8ZNzVGskbJLq6OONQuJU8AEkNxmRHkb39Kja21YOrfRfdN2Qc-CKo0Kmj_RVKSDReWr4B/s1600/Stats_et_al_chica_sprawl.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="715" data-original-width="796" height="287" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6e2LRxiYijblCyY3dGMQd23WmUXJtZHkiH3EDRZT55BomSfA5hLoU7QZLBg1nfwVIgLP5y4s8ZNzVGskbJLq6OONQuJU8AEkNxmRHkb39Kja21YOrfRfdN2Qc-CKo0Kmj_RVKSDReWr4B/s320/Stats_et_al_chica_sprawl.jpg" width="320" /></a></div>
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<i>
</i><div class="MsoNormal">
<br /></div>
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<br /></div>
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<br /></div>
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"The" and "a/an" are parts of speech
called articles. "The" is English's definite article, which means
it's used when there is a specific object being referred to, generally either
because only one exists, or because one example of the item is more important
than all the others.</div>
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<br /></div>
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"A" and "an" are indefinite articles.
They are used to refer to any one of multiple similar items. "An" is
used when the next word starts a with a vowel sound (A,E,I,O,U, but also HA,
HE, HI, HO, HU), and "a" is used for everything else. Special
attention is needed when referring to letters.</div>
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<br /></div>
<div class="MsoNormal">
"The restaurant got a D grade" (D is pronounced
"dee", which starts with a consonant)</div>
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<br /></div>
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"The restaurant got an F grade" (F is pronounced
"eff", which starts with a vowel)</div>
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<br /></div>
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---- </div>
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<br /></div>
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Setting the difference between "a" and
"an" is straight forward, but what about "a" versus
"the"? Consider:</div>
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<br /></div>
<div class="MsoNormal">
<br /></div>
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"Here is A glass of water" (This sentence implies
there are multiple glasses of water, and this one is just only one.)</div>
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<br /></div>
<div class="MsoNormal">
"Here is THE glass of water." (This implies that
there is only one relevant glass of water)</div>
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<br /></div>
<div class="MsoNormal">
"Here is THE glass of water you requested." (Does
not imply glasses of water in general. It does imply that you requested only
one glass.)</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>General rule:</b> When something is mentioned already,
there is only one relevant instance of that thing, so "the" is
used.<span style="mso-spacerun: yes;"> </span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
---- </div>
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<br /></div>
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<i>I ran a (1) regression tree analysis to find which
variables were important. The (2) analysis showed that "age" was the
(3) most important variable.</i></div>
<div class="MsoNormal">
<br /></div>
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For (1), many regression tree analyses could be done and
have been done, and this is just one of those many, so we use "a".</div>
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<br /></div>
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For (2), we're talking about the same analysis as in (1),
but at this point it's been established that there is one relevant analysis, so
we use "the".</div>
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<br /></div>
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For (3), there is only one 'most important variable', so we
use "the".</div>
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<br /></div>
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<br /></div>
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<b>General rule:</b> "the" can refer to multiple
objects, as long as it's all the objects in that collection.</div>
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<br /></div>
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<br /></div>
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<i>After generating a (1) random forest, we found the (2)
most important variables. Age was an (3) important variable, but so was income.</i></div>
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For (1), this random forest is just one of many that could
be generated.</div>
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<br /></div>
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For (2), although there are multiple variables being
referenced, we are referring to all of them, so we use "the" to help
show that there are no others. As a general rule</div>
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<br /></div>
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For (3), age is only one of the multiple important
variables, so we use "an".</div>
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<br /></div>
<div class="MsoNormal">
<b>General rule: If you want to refer to multiple objects,
but part of a collection, you can do so by prepending some indication that
you're only looking at a part.</b></div>
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<br /></div>
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---- </div>
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<br /></div>
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<i>Half of the (1) data points are positive, a few of the
(2) points are outliers.</i></div>
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<br /></div>
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For (1), "the data points" refers to all of the
points, and "Half of" modifies this.</div>
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<br /></div>
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For (2), "the points" is a shorter version of
"the data points", and "a few of" modifies this. We have
already established that they're data points, so we can shorten this to
"points".</div>
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<br /></div>
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----</div>
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<br /></div>
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<i>Most of the (1) temperature values are negative, but a
few of them (2) are above 30.</i></div>
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<br /></div>
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For (1), there are multiple temperature values, "the
temperature values" would refer to all of the values"; 'most of'
modifies that.</div>
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<br /></div>
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For (2), "them" is a shortcut to referring to
"temperature values", which have already established as subject of
the paragraph.</div>
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<br /></div>
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----</div>
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<br /></div>
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<i>Some of the (1) time, I wonder about rounding error.
Sometimes, I think they (2) are overlooked.</i></div>
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For (1), "the time" is an abstraction, and it
always needs a modification, even if that modifier is "all of". We
could shorten "Some of the time" to "Sometimes" like we did
in the second sentence.</div>
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<br /></div>
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For (2), "they" is a shortcut to referring to
"rounding error". "They" refers to a plural, and there is
only one plural it could refer to.</div>
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<br /></div>
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<b>General rule: </b>We can use "the" and
"a" to imply things that would not otherwise be obvious in the
writing.</div>
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<br /></div>
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---- </div>
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</div>
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<span style="mso-spacerun: yes;"> </span><i>The (1) woman who
wrote this (2) R package is a (3) genius.</i></div>
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<br /></div>
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For (1), "The woman" tells the reader that there
is only one woman who wrote this R package, even if it hasn't yet been
established who that is yet.</div>
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<br /></div>
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For (2), "this R package" assumes that the reader
knows what R package is being referred to.</div>
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<br /></div>
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For (3), "a genius" implies that more than one
genius exists, and the woman is one of them.</div>
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<br /></div>
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<i>---- </i></div>
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<br /></div>
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<i>In the (1) third step of the (2) optimization, the (3)
parameters are updated to a (4) better fit to the (5) data.</i></div>
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<br /></div>
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For (1), there may be multiple steps in the optimization,
but only one of them is the third one.</div>
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<br /></div>
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For (2), there may be many optimizations, but by the rest of
the context, it's clear that one particular optimization is being described. We
could also write "this optimization".</div>
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<br /></div>
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For (3), "the parameters" refers to all the
parameters being used.</div>
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<br /></div>
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For (4), "a better fit" implies that there could
be other better fits too. If it was the best fit, it has to be "the best
fit" instead of "a best fit" because "best" implies
only one.</div>
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<br /></div>
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For (5), "the data" refers to all of the relevant
data.</div>
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<br /></div>
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<br /></div>
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<b>General rule:</b> If something is obviously unique, or if
something is being referred to in the abstract, usually we don't need
"the" or "a".</div>
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<br /></div>
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<i> -----</i></div>
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<br /></div>
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<i>Linear regression (1) is an (2) unbiased method to fit
data(3), therefore the (4) mean of the (5) residuals is zero.</i></div>
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<br /></div>
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For (1), "Linear regression" is a name for a
specific method, it's already clear that there is only one method called
"linear regression" without using "the".</div>
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<br /></div>
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For (2), there is more than one "unbiased method",
so use "an".</div>
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<br /></div>
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For (3), "data" is being referred abstractly. If we
were referring to some specific set of data, we use either "a set of
data" (because there are many sets, this is only one), or "the
data" (because, while there is many points of data, we are referring to
all them in a set).</div>
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<br /></div>
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For (4), there is only one mean of these residuals.</div>
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<br /></div>
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For (5), all of the results are being referred to.</div>
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<br /></div>
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<br /></div>
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---- </div>
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<br /></div>
<div class="MsoNormal">
<i>Google (1) used to recruit employees (2) by anonymously
posting puzzles in public locations (3).</i></div>
<div class="MsoNormal">
For (1), "Google" is a proper name for a company,
so we don't need to clarify that it's unique.</div>
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<br /></div>
<div class="MsoNormal">
For (2), it's already clear that "employees" is
referring to Google's employees, so, while we could say "Google's
employees" or "their employees", we don't need to.</div>
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<br /></div>
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---- </div>
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<br /></div>
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<i>New York (1) is the city (2) that hosts IBM's main
headquarters.</i></div>
<div class="MsoNormal">
For (1), "New York" (or "New York City")
is a proper name of a city. Even if they are multiple places called New York,
we wouldn't say "a New York", we would clarify with additional
information like the US state (New York, New York).</div>
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<br /></div>
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For (2), there is only one city that hosts IBM's main HQ. We
write "the" because we're now adding new information beyond simply
the name of the city.</div>
<div class="MsoNormal">
(A much cleaner way to write this whole sentence would be
"New York hosts IBM's main headquarters")</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Some more specific examples and rules can be found in "The
definite article" by EF Education First</div>
<div class="MsoNormal">
<a href="https://www.ef.com/wwen/english-resources/english-grammar/definite-article/">https://www.ef.com/wwen/english-resources/english-grammar/definite-article/</a></div>
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<br /></div>
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<br /></div>
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<br /></div>
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<b>Mini exercise: Fix the following sentences.</b></div>
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<i>(Answers after the tiger lily) </i><b><br /></b></div>
<div class="MsoNormal">
<br /></div>
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Q1: It's not possible to describe whole story</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q2: Instead of training a simple model directly from
training set,</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q3: simple model outperforms the same model trained directly
from training set.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q4: On each leaf node, samples are split into training set
and testing set.</div>
<div class="MsoNormal">
<br /></div>
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Q5: Figure 1 shows that false positive rate is larger.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q6: We used the string-manipulation functions in stringr
package.</div>
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<br /></div>
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Q7: proof of this theorem is found in appendix.</div>
<div class="MsoNormal">
<br /></div>
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Q8: In next section, we discuss alternate to our method.</div>
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<br /></div>
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Q9: By equation (1), together with above two implied
integrals, we have a complete proof.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q10: As a result, new method selects very few explanatory
variables, which causes poor model fits.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q11: ensemble approach is based on combination of many
models.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q12: Kernel density estimation is based on mean density of
smoothed individual observations.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q13: We assume that there is no missing data in each of
datasets.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q14: Consider simplified problem, in which N=5.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q15: Consider simplest problem, in which N=1.</div>
<div class="MsoNormal">
<br /></div>
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Q16: Note that the minimum value for the test statistic is
zero, and that upper bound does not exist.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q17: Thus, only marginal distribution is known.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Q18: We obtain the following mixture of k densities as the
aggregate density for k-th class under the </div>
<div class="MsoNormal">
following</div>
<div class="MsoNormal">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM2XXtjpZ0jdBqQTOId-rWSFGTwRHEWcN6Yy9biTWDl_QnyM-f2W3RkGOocubIS-65mSarTpWfEEAT2UroOsCZyFl7rRP1Q0ll7NmPnpJO4iTVyLE4K-fy6uRRTAP4EFfqgtVd43DSz26y/s1600/stats-et-al+tiger+lily.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="791" data-original-width="1224" height="206" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM2XXtjpZ0jdBqQTOId-rWSFGTwRHEWcN6Yy9biTWDl_QnyM-f2W3RkGOocubIS-65mSarTpWfEEAT2UroOsCZyFl7rRP1Q0ll7NmPnpJO4iTVyLE4K-fy6uRRTAP4EFfqgtVd43DSz26y/s320/stats-et-al+tiger+lily.jpg" width="320" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
A1: It's not possible to describe [THE] whole story</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
A2: Instead of training a simple <span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">model directly from [A] training set,</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A3: [THE] simple model outperforms the same model trained directly
from [A] training set.</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A4: On each leaf node, samples are split into [A] training set and [A]
testing set.</span></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A5: Figure 1 shows that [THE] false
positive rate is larger.</span></div>
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"> </span><div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A6: We used the string-manipulation
functions in [THE] stringr package.</span></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A7: [A] proof of this theorem is found
in [THE] appendix.</span></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A8: In [THE] next section, we discuss
[AN] alternate to our method.</span></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A9: By equation (1), together with
[THE] above two implied integrals, we have a complete proof.</span></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A10: As a result, [THE] new method
selects very few explanatory variables, which causes poor model fits.</span></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A11: [THE] ensemble approach is based
on [A] combination of many models.</span></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A12: Kernel density estimation is
based on [THE] mean density of [THE] smoothed individual observations.</span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A13: We assume that there is no
missing data in each of [THE] datasets.</span></div>
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<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A14: Consider [A] simplified problem,
in which N=5.</span></div>
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<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A15: Consider [THE] simplest problem,
in which N=1.</span></div>
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<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A16: Note that the minimum value for
the test statistic is zero, and that [AN] upper bound does not exist.</span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: 200%; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A17: Thus, only [THE] marginal
distribution is known.</span></div>
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<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">A18: We obtain the following mixture
of k densities as the aggregate density for [THE] k-th class under the
following conditions:</span></div>
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<br /></div>
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<span style="mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">---- </span></div>
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<br /></div>
Scientific writing doesn't get the attention it deserves, but the work of <a href="https://sandylittletree.net/">Dr. Sandy Littletree</a>'s Indigenous Library and Information Services deserves a lot more attention. Library work is more than just shelving books, it's a hefty task to organize the world's information, double so for indigenous info. Go 'check out' <a href="https://sandylittletree.net/">Dr. Littletree's projects here!</a><!--[if gte mso 9]><xml>
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<![endif]-->Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-3669703689880260352020-06-13T23:00:00.000-07:002020-06-13T23:00:25.446-07:00R Packette - Fraction Matrix Operations
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Open up any linear algebra textbook and have a look at the matrix
entries. Are there all integers? Are they all written as decimals? There's
probably at least some that are fractions. Matrices in computer programs are
almost always in decimal form. The exception is symbolic mathematics programs
like Maple and Mathematica. That's because computers store non-whole numbers as
floating-point values instead.</div>
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Floating-points are fine most of the time, but they're often
not exact. What happens if we work the fractions directly?</div>
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The fractions 1/2, 1/4, and 3/4 are represented as 0.1,
0.01, and 0.11 in binary, respectively, but 1/3 and 1/5 are repeating values
that can't be represented exactly. The fraction 1/3 is 0.0101010101… and 1/5 is
0.001100110011.. <span style="mso-spacerun: yes;"> </span>, so when using these numbers,
precision is lost just by storing the number; it can only get worse with calculations. <br />
<br />
From there, any calculation using that number is going to propagate and
accumulate. With a fraction of integers, you can maintain exactitude longer.
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The other possible advantage is speed, (although not in my
implementation, it's R instead of C or assembly code). Computers can add,
subtract, and multiply numbers in binary very efficiently. However, binary
division has to be done with long division* , which is N times slower, where N
is the number of bits of accuracy.</div>
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But with fractions, division is just cross-multiplication,
which is two quick multiplications instead of one slow division.</div>
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I wrote a set of R functions that handles some common matrix
operations for matrices of fractions instead of floating-point numbers. Only in
edge cases are these preferable than the existing functions, but </div>
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The set of R functions in this source file in this <a href="https://drive.google.com/drive/folders/1XwqVFnr7fylL3O3QGdJwLkrz2NWgXQJH?usp=sharing">google docs folder</a> are a proof of concept. They perform some scalar and matrix operations by
inputting a fraction, outputting a fraction, and at no internal point relying
on floating-point arithmetic. </div>
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This isn't trivial. Consider the inner product of
two vectors of length n:</div>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="footer"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="index heading"/>
<w:LsdException Locked="false" Priority="35" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="caption"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="table of figures"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="envelope address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="envelope return"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="footnote reference"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation reference"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="line number"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="page number"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="endnote reference"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="endnote text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="table of authorities"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="macro"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="toa heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 5"/>
<w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Closing"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Signature"/>
<w:LsdException Locked="false" Priority="1" SemiHidden="true"
UnhideWhenUsed="true" Name="Default Paragraph Font"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Message Header"/>
<w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Salutation"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Date"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Block Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
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This formula is used in the function <span style="font-family: "Courier New";">innerprod.fracvec()</span>, which takes two vectors of fractions
and returns a single fraction. </div>
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<br /></div>
<div class="MsoNormal">
Alternatively, one could rewrite matrix operations in terms
of scalar fraction operations instead of trying to do everything in one big
formula, but there's a loss of speed in doing basic arithmetic as function
calls. This is what I ended up doing for <span style="font-family: "Courier New";">fracmat.upper.tri()</span>
the function that converts a square matrix into an upper triangular form.<span style="font-family: "Calibri",sans-serif; font-size: 11.0pt; line-height: 107%; mso-ansi-language: EN-CA; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;"> </span></div>
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<br /></div>
<span style="font-family: "Calibri",sans-serif; font-size: 11.0pt; line-height: 107%; mso-ansi-language: EN-CA; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: "Times New Roman"; mso-bidi-language: AR-SA; mso-bidi-theme-font: minor-bidi; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;">Here's a few test cases selected from the test
case file in the</span> <a href="https://drive.google.com/drive/folders/1XwqVFnr7fylL3O3QGdJwLkrz2NWgXQJH?usp=sharing">google docs folder</a><br />
<br />
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="index 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="index 9"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 2"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 3"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 5"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 6"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 7"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 8"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 9"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="header"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="35" SemiHidden="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="envelope address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="toa heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="1" SemiHidden="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
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<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
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<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
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<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
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<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
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<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
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<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
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<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
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<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
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<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
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<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
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<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
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<br />
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">## Case 1</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"># Matrix
multiplication, floating point vs fraction matrix</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"># Make some
random matrices of fractions, and get their product</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">A =
matrix(sample(1:9),nrow=3,ncol=3)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">B =
matrix(sample(1:9),nrow=3,ncol=3)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">F =
fracmat(A,B)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">C =
matrix(sample(1:9),nrow=3,ncol=3)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">D =
matrix(sample(1:9),nrow=3,ncol=3)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">G =
fracmat(C,D)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">F_float =
A/B</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">G_float =
C/D</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">prod_float
= F_float %*% G_float</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">FG =
multiply.fracmat(F,G)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">FG</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">$`top`</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1,]<span style="mso-spacerun: yes;"> </span>143<span style="mso-spacerun: yes;">
</span>97<span style="mso-spacerun: yes;"> </span>577</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[2,]
2837<span style="mso-spacerun: yes;"> </span>23<span style="mso-spacerun: yes;"> </span>59</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[3,]<span style="mso-spacerun: yes;"> </span>172<span style="mso-spacerun: yes;">
</span>29<span style="mso-spacerun: yes;"> </span>61</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">$bottom</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1,]<span style="mso-spacerun: yes;"> </span>36<span style="mso-spacerun: yes;">
</span>42<span style="mso-spacerun: yes;"> </span>126</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[2,]<span style="mso-spacerun: yes;"> </span>420<span style="mso-spacerun: yes;">
</span>15<span style="mso-spacerun: yes;"> </span>15</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[3,]<span style="mso-spacerun: yes;"> </span>7<span style="mso-spacerun: yes;">
</span>4<span style="mso-spacerun: yes;"> </span>4</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">prod_frac =
FG$top / FG$bottom</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">prod_float</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>[,1]<span style="mso-spacerun: yes;"> </span>[,2]<span style="mso-spacerun: yes;">
</span>[,3]</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1,]<span style="mso-spacerun: yes;"> </span>3.972222 2.309524<span style="mso-spacerun: yes;"> </span>4.579365</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[2,]<span style="mso-spacerun: yes;"> </span>6.754762 1.533333<span style="mso-spacerun: yes;"> </span>3.933333</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[3,]
24.571429 7.250000 15.250000</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">prod_frac</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>[,1]<span style="mso-spacerun: yes;"> </span>[,2]<span style="mso-spacerun: yes;">
</span>[,3]</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1,]<span style="mso-spacerun: yes;"> </span>3.972222 2.309524<span style="mso-spacerun: yes;"> </span>4.579365</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[2,]<span style="mso-spacerun: yes;"> </span>6.754762 1.533333<span style="mso-spacerun: yes;"> </span>3.933333</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[3,]
24.571429 7.250000 15.250000</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">sum(abs(prod_float
- prod_frac))</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1]
1.554312e-15</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">### Case 2</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">## Get the
inverse and determinant of a random matrix</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">## Compare to
the answers from base::det and base::solve</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">A =
matrix(sample(1:20)[1:9],nrow=3,ncol=3)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">B =
matrix(sample(1:20)[1:9],nrow=3,ncol=3)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">A</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1,]<span style="mso-spacerun: yes;"> </span>20<span style="mso-spacerun: yes;">
</span>18<span style="mso-spacerun: yes;"> </span>3</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[2,]<span style="mso-spacerun: yes;"> </span>2<span style="mso-spacerun: yes;">
</span>16<span style="mso-spacerun: yes;"> </span>8</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[3,]<span style="mso-spacerun: yes;"> </span>4<span style="mso-spacerun: yes;">
</span>10<span style="mso-spacerun: yes;"> </span>7</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">B</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>[,1] [,2] [,3]</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1,]<span style="mso-spacerun: yes;"> </span>8<span style="mso-spacerun: yes;">
</span>9<span style="mso-spacerun: yes;"> </span>17</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[2,]<span style="mso-spacerun: yes;"> </span>10<span style="mso-spacerun: yes;">
</span>1<span style="mso-spacerun: yes;"> </span>18</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[3,]<span style="mso-spacerun: yes;"> </span>12<span style="mso-spacerun: yes;">
</span>14<span style="mso-spacerun: yes;"> </span>5 </span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">F =
fracmat(A,B) #Fraction matrix object</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">F_float =
A/B #Floating point matrix</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"># Get the
determinants, compare.</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">fracmat.det(F)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">$`top`</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1]
8592592140</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">$bottom</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1]
159043500</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">fracmat.det(F)$top
/ fracmat.det(F)$bottom</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1]
54.02668</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">det(F_float)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1]
54.02668</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"># Get the
inverse of each, display, and compare</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">Finv1 =
fracmat.inverse(F)$top / fracmat.inverse(F)$bottom</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">Finv2 =
solve(F_float)</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">Finv1</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>[,1]<span style="mso-spacerun: yes;"> </span>[,2]<span style="mso-spacerun: yes;"> </span>[,3]</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1,]<span style="mso-spacerun: yes;"> </span>0.408733982 -0.04949313 -0.03580898</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[2,]
-0.002440495<span style="mso-spacerun: yes;"> </span>0.06369402 -0.01991270</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[3,]
-0.096072464 -0.02071287<span style="mso-spacerun: yes;"> </span>0.73297120</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">Finv2</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>[,1]<span style="mso-spacerun: yes;"> </span>[,2]<span style="mso-spacerun: yes;"> </span>[,3]</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1,]<span style="mso-spacerun: yes;"> </span>0.408733982 -0.04949313 -0.03580898</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[2,]
-0.002440495<span style="mso-spacerun: yes;"> </span>0.06369402 -0.01991270</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[3,]
-0.096072464 -0.02071287<span style="mso-spacerun: yes;"> </span>0.73297120</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">sum(abs(Finv1
- Finv2))</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Courier New"; font-size: 10.0pt; line-height: 107%;">[1]
1.065684e-12</span></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal">
Here's a catalogue of the functions included in this
packette. The source file has more detailed comments explaining each one.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
These functions establish a new fraction matrix, vector, or
value from a pair of matrices, vectors, or scalars of equal dimensions
respectively (top, bottom). When <span style="font-family: "Courier New";">enforce_integer
= TRUE</span>, the function rounds each entry of the top and bottom parts. The
function <span style="font-family: "Courier New";">fraction()</span> is much
simpler and doesn't include rounding or quality checks because it's only
intended for internal use.</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fracmat(top=NA,
bottom=NA, enforce_integer=TRUE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fracvec(top=NA,
bottom=NA, enforce_integer=TRUE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fraction(A,B)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
These functions convert a fraction matrix or vector back
into a matrix or vector of decimal (floating-point) numbers.</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fracvec.to.vec(fv)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fracmat.to.mat(fm)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
There are shortcuts to produce commonly used fraction
matrices</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">allsame.fracmat(top,
bottom=1, nrow, ncol)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">allzero.fracmat(nrow,
ncol)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">allone.fracmat(nrow,
ncol)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">identity.fracmat(size)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Many of the following functions have an argument called
'simplify', which defaults to true. These are functions which call
simplify.fracmat, which divides the numerator and the denominator of each
fraction by their greatest common denominator.</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">simplify.fracmat(fm,
nPrimes=10, report_overflow = FALSE, fuzzy_reduce = TRUE, maxint = 10^15)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
If any part of the fraction matrix is larger than maxint,
which is close to the limit of the modulo function, fuzzy_reduce will scale the
fraction in question down so the larger of the numerator and denominator is
maxint.</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fuzzy_reduce(top,
bottom, limit = 10^15)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
These do elementwise addition and subtraction of fractions
in two fraction matrices or vectors.</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">add.fracmat(fracmatAB,
fracmatCD, subtraction = FALSE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">subtract.fracmat(fracmatAB,
fracmatCD)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
This does matrix multiplication of fraction matrices. It does
the by iteratively calling the inner product function on rows of the left matrix
(fracmatAB) and columns of the right matrix (fracmatCD)</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">multiply.fracmat(fracmatAB,
fracmatCD, simplify=TRUE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">innerprod.fracvec(fracvecAB,
fracvecCD, simplify=TRUE)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Just calling a fracmat object will give you two matrices,
but <span style="font-family: "Courier New";">print.fracmat()</span> will output
text of each fraction with the numerator and denominator together.</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">print.fracmat(fm)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
This function converts a single matrix of floats (floatmat)
to a fraction matrix. It sets the denominators to 10^max_digits and the
numerator to floatmat*10^max_digits (it will use fewer digits if exactitude is
reached)</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">mat.to.fracmat(floatmat,
max_digits = 6, simplify = TRUE)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="mso-tab-count: 1;"> </span></div>
<div class="MsoNormal">
Does the row operations on a fraction matrix fm to turn it
into an upper triangular form</div>
<div class="MsoNormal">
Returns the upper triangular matrix U and 'inverse', which
records the same row operations starting from the identity</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fracmat.upper.tri(fm,
track = FALSE, output_inverse = TRUE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fracmat.upper.tri.to.ident(fmU,
inv_fm, track = FALSE)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
These get the determinant and the inverse of a fraction
matrix. They do so with calls to <span style="font-family: "Courier New";">fracmat.upper.tri
and fracmat.upper.tri.to.ident</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fracmat.det(m,
track=FALSE, output_float = FALSE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">fracmat.inverse(m,
track=FALSE)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
There is also a decimal (floating-point) matrix version of
these high-level functions, which are useful for developing and testing the
fraction matrix counterparts. <span style="font-family: "Courier New";">float.inverse</span>
and <span style="font-family: "Courier New";">float.det</span> should behave
identically to how <span style="font-family: "Courier New";">base::solve()</span>
and <span style="font-family: "Courier New";">base::det()</span> on single square
matrices.</div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">float.det(m,
track=FALSE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">float.inverse(m,
track=FALSE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">float.upper.tri(m,
track=FALSE, output_inverse =TRUE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">float.upper.tri.to.ident(U,
inv_m, track=FALSE)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
These perform elementary fraction arithmetic functions. They
are used internally for row-operation functions like <span style="font-family: "Courier New";">fracmat.upper.tri</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">negate.fraction(fm,
negate_denom = FALSE)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">add.fraction(fracAB,
fracCD)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">subtract.fraction(fracAB,
fracCD)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">multiply.fraction(fracAB,
fracCD)</span></div>
<div class="MsoNormal">
<span style="font-family: "Courier New";">divide.fraction(fracAB,
fracCD)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
If I come back to this, this is my wish list:</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>General development:</b></div>
<b>
</b><div class="MsoNormal">
- More unit testing, stress testing, speed testing.</div>
<div class="MsoNormal">
- Further development into a bona-fide package.</div>
<div class="MsoNormal">
- Is any of this paper-worthy?</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Additional matrix functions:</b></div>
<b>
</b><div class="MsoNormal">
- Recording of row operations.</div>
<div class="MsoNormal">
- LU decomposition</div>
<div class="MsoNormal">
<span style="mso-tab-count: 1;"> </span></div>
<div class="MsoNormal">
<b>Additional presentation:</b></div>
<b>
</b><div class="MsoNormal">
- Print out records of matrices after row operations in a
way that Linear Algebra students can use </div>
<div class="MsoNormal">
- Print fraction matrices in latex code</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Avenues for improved fraction simplification:</b></div>
<b>
</b><div class="MsoNormal">
- Integration with the big64 package so 64-bit integers can
be handed exactly.</div>
<div class="MsoNormal">
- Possible integration from the VeryLargeIntegers so arbitrarily large integers can be handled, however since VeryLargeIntegers
operates by string manipulation, it is much slower than floating point arithmetic. </div>
<div class="MsoNormal">
- Possibly passing the whole thing over to RcppBigIntAlgos
which is designed to factor large integers.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
But what do you think? I am open to suggestion at
jackd@sfu.ca , jack.davis@sportlogiq.com , or <a href="https://twitter.com/jack_davis_sfu">@jack_davis_sfu</a> on Twitter.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
* (or multiplication by an inverse, or subtraction of
logarithms, which are all either slower or much more memory intensive)</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
References: <u>Linear Algebra: An interactive approach</u>
by Jain and Gunawardena, especially the worked example on page 118 for the detailed test case.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<u>The End of Error: Unum Computing</u> by John L. Gustafson
for its alternative approaches to rounding error. </div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Shout out to <a href="https://twitter.com/That_Astro_Chic">Ashley Lindalia</a> the astrochemist, because 'astrochemist' is an amazing title and because planetary science matters more than anything I do.</div>
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Name="footnote reference"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation reference"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="line number"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="page number"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="endnote reference"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="endnote text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="table of authorities"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="macro"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="toa heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Bullet 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number 5"/>
<w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Closing"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Signature"/>
<w:LsdException Locked="false" Priority="1" SemiHidden="true"
UnhideWhenUsed="true" Name="Default Paragraph Font"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Message Header"/>
<w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Salutation"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Date"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Block Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
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<![endif]-->Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-56166809594808761132020-02-28T15:57:00.000-08:002020-07-27T12:16:22.751-07:00First impressions of the new XFL<div class="separator" style="clear: both; text-align: center;">
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In 2001, the XFL, short for the X Football League* started as a sort of college-level alternate to the NHL, the premier gridiron football league in the United States. Establishing a new team is hard, let alone an entirely new league, and the XFL shut down after only one season and a ton of controversy.<br /><br />In 2020, somehow, it came back.<br />
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<a name='more'></a><br />I watched a recorded XFL game from a few days prior (Dallas Renegades at Seattle Dragons Feb 22, 2020), and here were my thoughts as they happened.<br /><br />First, the tv production value is nearly as good, or as good, as an NFL game. They're playing in Seattle's NFL field, so all the rigging and camera equipment is still there. That TV equipment would otherwise be unused. An NFL game costs over 10 million dollars to televise and a lot of that goes to all this absurd rigging and camera stuff that they can only use 8-12 times per stadium per year. I imagine the XFL is getting comparable coverage for a small fraction of the price.<br /><br />The stadium is half full. They only sold the lower deck seats. A quick check shows that typical tickets cost $20 USD for this game. That's in comparison to $200+ for the NFL Seattle Seahawks that were playing here just 3 weeks earlier.<br /><br />They're playing in late winter, early spring after the NFL season and before the stadium will be used a lot for concerts. NBA isn't in playoffs yet, but most of teams are decided already. MLB is only in spring training. To a typical football fan there isn't much else to watch. This is smart business all around.<br /><br />At one point, the commentators said this crowd of 29,000 was the highest yet of the 3 weeks that the league has been running. Another crowd elsewhere today was 22,000. Those numbers are smaller than most CFL (Canadian Football League) games, but not by much. That is, except for the Saskatchewan Roughriders, who regularly sell out in a stadium that can fit 10 percent of the province's population.<br /><br />Let's talk the game experience itself. I'm using the skip button a lot less than normal for a football game because there's less to skip. There's less down time between plays.<br /><br />They're using the same style graphical overlays and animations that they use in NFL games, but with different logos. It's like they're really trying to show themselves as an alternative, not a lesser league. <br /><br />In NFL, CFL, and NCAA college bowl, the kicking team all start way over to the left and have all that space to pick up steam and crash into each other at full speed. It also takes a lot of time and the kickoff return rarely strays from the default 25-yard line. Jon Bois did a convincing video on kickoffs being stupid and dangerous.<br /><br />But here, kickoffs are fast, statistically meaningful, and designed to minimize brain trauma. Refer to this terrible phone photo. The kicker is way over to the left and has to kick over the two lines of players to a receiver way over the right. There's no full-speed crashes, and the outcome of the return doesn't feel like a forgone conclusion.<br /><br /><br />
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<tr><td class="tr-caption" style="text-align: center;">XFL Kickoff and Return (Stats-et-al.com)</td></tr>
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<br />Speaking of forgone conclusions, there's a new extra point structure for after touchdowns. First, no more kicking for extra points. You have to run or pass it into the end zone for the extra points every time. Also, the farther out you start from, the more extra points you can get: one, two, or three points based on the line of scrimmage you choose. (This part feels a little gimmicky, but not as bad as the North Korean Basketball's three-point dunks and EIGHT POINTS for scoring in the last three seconds of the game). A scorigami is a final game score that has never happened in a league before, and looking XFL scorigami could be flexible extra point structure.<br /><br />There were five touchdowns and one field goal in this game. From those five touchdowns, every attempt was from the two-point line and extra attempt failed. Late in the game, the commentators mentioned how either going for 1 or 3 would be smarter because given that nobody has gotten any extra points, 1 point is like a tiebreaker for touchdowns, and 3 points is like getting a whole extra field goal. It's going to take some time and data before the nerds figure out the optimal strategy on this, and if this league sticks, it's going to start a lot of bar-room arguments, which is also good for business. <br /><br />There's a penalty or restriction related to punting I don't understand. However, it must matter a lot because there's few punts and even some turnovers from not getting 10 yards in four downs.<br /><br /><br />
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<tr><td class="tr-caption" style="text-align: center;">XFL Extra Point (Stats-et-al.com)</td></tr>
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<br /><br />The players are not terrible. They're making a few mistakes but I don't know enough about football to know or care if this is unusual. It feels like there's lot of false starts and other flags thrown for things that can mostly be attributed to disorganization.<br /><br />The players they're choosing to highlight are CFL and NFL veterans. Given that NFL General Managers are letting go of players after only 2 years to save money, that probably means the XFL talent pool is probably healthy given that a lot of these players are viable, just not superstars worthy of a re-signing in the NFL.<br /><br />Nobody has been carried off in a stretcher, so that's a big plus. (Injuries are the main reason I don't watch football) One guy got a concussion but he walked unaided off the field after a minute. Another guy was holding his knee and they went to commercial break before I saw the outcome, but the play wasn't something that made me want to sympathy vomit.<br /><br />Also absent is Duane "The Rock" Johnson, or any other wrestling personality outside of commercial breaks. The XFL really has decided that sports are serious business. Halftime show? No halftime show. Ten minute break and back to work!<br /><br />Well, not completely serious. Look at this fan.<br /><br />(XFL_dragonhead.jpg)<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbFg-P6WzU9AO1zOweDE3LXKlipTRWVYi4HDhOXAfDPfLpLFIV8BhG0q_tmlcF26vLm_bOLrvFWEde1oPwIAIVijWiajpyt43CG_FcsL41kCgp_R9a1Yf-TKaM6VNx-Nuq90G44UhCMwfi/s1600/XFL_dragonhead.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1600" data-original-width="1215" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbFg-P6WzU9AO1zOweDE3LXKlipTRWVYi4HDhOXAfDPfLpLFIV8BhG0q_tmlcF26vLm_bOLrvFWEde1oPwIAIVijWiajpyt43CG_FcsL41kCgp_R9a1Yf-TKaM6VNx-Nuq90G44UhCMwfi/s320/XFL_dragonhead.jpg" width="242" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">XFL Dragonhead fan (Stats-et-al.com)</td></tr>
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<br /><br />That is not an official merchandise helmet, right? There's a whole bunch of different helmets and handmade dragon wings and such in the crowd. Naming your team the dragons is just an invitation for custom cosplay, especially in the city with Wizards of the Coast HQ.<br /><br />The game is over now: Dallas 24, Seattle 12. It was pretty close until the last ten minutes. It's the post-game and the broadcasters are showing the point spread and over-under betting lines right there on TV, so I guess they're not shy about gambling. That's unusual for North American sports, at least on a game's mainstream broadcast. For any bookmakers reading this: scorigami prop bets, think about it, and call me if you want to make it happen.<br /><br /><br />I'm not a football fan. I probably never will be, but this has managed to keep my curiosity enough to consider watching a second game. There are people (mascots?) in green dragon suits on the field. Are they mascots or just more furries? The dragons lost this game, yet the suited people are dancing around. I remain uncertain.<br /><br />In the first iteration, the XFL was heavily advertised before launch, and was sold as a sort of World Wrestling Entertainment extension with cheerleaders and maybe some college football in between.<br /><br />This time it feels a lot more like an actual sport. Instead of flashy gimmicks, the XFL website directs you to the list of rule changes and mechanics that make them different from other leagues, with explanations that sound like they come a league commissioner updating a rule book.<br /><br />But the XFL has already won in my book because it doesn't have convicted dog fighter Michael Vick.<br /><br /><br /><i>*That's right, the X in XFL doesn't stand for extreme or xtreme or x-rated, or extra or... well... anything. It just is. Source: Brett Forrest's book, "Long Bomb: How the XFL Became TV's Biggest Fiasco"</i>Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-62706098787957724842019-11-09T16:05:00.001-08:002019-11-09T16:05:47.680-08:00Basketball Data Science with Applications in R - An Advance ReviewMy overall impression of "Basketball Data Science" is that it's exactly the sort of book I would recommend to an instructor or able student of statistics in sport. Most of my criticisms are because it's not the book I imagine I would have written, not because the book isn't good.<br />
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Transparency statement: </h4>
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This is a review of an advance beta copy of Basketball Data Science by Paola Zuccolotto and Marica Manisera that I received from CRC Press as one of 9 people to determine the market viability this book. I was paid to review this book, but I was not paid to publish that review. I've done my best to update this review to reflect the changes made to the book since then.<br />
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General impression:</h4>
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On a spectrum from "all methodology", like Casella and Berger's famous textbook, to "all results" like a typical sports almanac, I would put "Basketball Data Science" between Casella-Berger and Jim Albert's "Visualizing Baseball". "Basketball Data Science" is intended to show how the data analysis is done, rather than revealing many sports-related insights directly. Also, this is a data science book, not a classical statistics book, and it focuses much more on description than on testing hypotheses. By my count, there are only 4 p-values used in all 257 pages.<br />
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This is mostly, but not entirely, the right choice. In one case, the authors claim that the Golden State Warriors tend to concentrate their shots in the first half of each quarter and first half of each game, effectively claiming that the team focuses 52% and 54% of their shots, respectively, in 50% of the game time. <br />
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Each chapter goes through a different type of analysis, including basic player statistics, passing tendencies, pace-of-play comparisons, and spatial analysis. One chapter looks at the inequality of points scored, with tools like the Lorenz curve and the Gini index which can be readily applied to other sports for things like shots, passes, and time in play.<br />
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The book reads like an extended manual for the author's R package, BasketballAnalyzeR, which is good for cohesive, digestible study material that minimizes external dependencies. However, this isn't the only R package for examining NBA data available. The package "py_ball", for example, is a Python API wrapper for both the NBA and WNBA websites. This isn't an oversight, but a deliberate choice by the authors for the sake of accessibility. It also minimizes dependencies on external sources and software.<br />
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On that note, I was disappointed to find that the book only included analysis and information related to the NBA. Along with the WNBA, there are leagues outside of North America that could really use some more statistical attention as well. (The authors addressed this by saying that they had tried to get data from the WNBA as well, but that it wasn't available. Yes, this is hypocritical of me to complain about given my own work.)<br />
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Specifics:</h4>
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The book uses radial plots to demonstrate disparate statistical features about a player. I like them. The big issue with radial / umbrella plots is that they're sensitive to the order in which you arrange statistics, and I think the order chosen here is well thought out. Disagreements and hate mail welcome.<br />
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Early in the book, the authors mention that range of shots is not a reliable measure in the example case because of a single outlier player, Steven Adams. There's a teaching opportunity here for any instructors willing to follow the footnotes and references: What do you do in the face of unusual or unexpected data?<br />
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For the inequality section, a possible research question for students could address how the inequality of passes compares with time spent on the court per player, or the time shared between two players on the court? Do any of your conclusions change if you use a different measure of inequality like the Shannon diversity index?<br />
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It was surprising to learn that almost every player falls into one of three positions (guard, forward, and center), especially given how fluid the movement seems and how much the paradigms of basketball have changed over time. I specifically asked about this the first version of this review, and the authors added a footnote about hybrid positions "such as the point forward (a hybrid PG/SF), the swingman (a hybrid SG/SF), the big (a hybrid C/PF) and the stretch four (a PF with the shooting pattern of a typical SG)." but all of these were apparently recognized in the 1950's and not since. I'd like to add one more special case to that list: Bob McAdoo, which Jon Bois explains at the 13:15 mark of The Bob Emergency: Part 2 https://youtu.be/dcGamPqUIxI?t=795 better than I ever could.<br />
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While fouls are mentioned a few times, and are readily available in the software package, there isn't much in the way of analysis of them. What happens when a star player has a lot of fouls? There's another student project.<br />
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To reiterate, I learned a lot from reading this, as a statistician and a relative outsider to basketball. It made the recent playoff games with the Toronto Raptors and the Golden State Warriors all the more enjoyable.Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0tag:blogger.com,1999:blog-5718736365000160223.post-85635041779028088442019-11-08T23:18:00.001-08:002019-11-08T23:18:13.269-08:00Postmortems from 'Game Developer', a book review<!--[if gte mso 9]><xml>
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UnhideWhenUsed="true" QFormat="true" Name="heading 8"/>
<w:LsdException Locked="false" Priority="9" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="heading 9"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="index 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="index 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="index 9"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
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<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 2"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 3"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 4"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 5"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 6"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 7"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 8"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" Name="toc 9"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="footnote text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="header"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="footer"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="index heading"/>
<w:LsdException Locked="false" Priority="35" SemiHidden="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="table of figures"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="envelope address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="footnote reference"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="endnote reference"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="endnote text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="table of authorities"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="toa heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Number"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="10" QFormat="true" Name="Title"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Signature"/>
<w:LsdException Locked="false" Priority="1" SemiHidden="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="39" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
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<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
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<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
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<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
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<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
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<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
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<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
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<br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">"Postmortems
from 'Game Developer'", edited by Austin Grossman is a collection of
after-the-fact analyses of popular video games that had been recently developed
and sold. The analyses are all written by senior members of each game's
development team, and usually the game's head creator. "Postmortems"
was published in 2003, so the games being analyzed include Diablo 2, Age of
Empires, System Shock 2, and Black and White. </span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">I
purchased my copy from a thrift store in 2016, which had a suggested Canadian
price of $42. This copy had several discount stickers on it, one of which had
the book for sale for $0.84, a 98% discount. </span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;"></span></div>
<a name='more'></a>Part of
this is probably owing to the book's cover, which makes it look like one of
those 'how to use technology' books that's obsolete before it hits the shelves,
like "Windows 95 for dummies", or "programming for IOS
8.2". But the content of the book isn't tied to any one technology, it's
about the principles of game design, has that much really changed?<br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">To a 2019
reader, there are two things that are notably absent. First, there's no modern
software development business lingo. Someone could read this book without any
knowledge of Agile. Instead of sprints, there was crunch time. Instead of, uh,
kanbans, there were to-do lists. That's not to say these methods weren't there
in other forms. In the postmortem of Myst 3: Exile, there was a technical
limitation during late development - the game took over an hour to
compile, and thus over an hour to test any change live. They adopted a buddy
system in which one programmer would watch another as they wrote code to do
quality assurance and minimize the number of test compiles necessary. That
buddy system today would be called "extreme programming". Buzzwords
ruin everything.</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">More
importantly, there's no almost discussion of revenue models. In 2003, most
games were sold by the copy, and that was the only time that revenue was made
from the game. In a couple of these games, there is some talk about preparation
for expansion packs, both free and paid. Only one game, "Dark Age of
Camelot" had a recurring revenue model in the form of a monthly
subscription. Card packs, microtransactions, energy systems, and loot boxes are
all nonexistent.</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">What was
surprisingly similar was the scale of the games. Each postmortem included a
development time and budget, and many of these games took 2+ years and $2-5
million to create. For AAA titles, that's not terribly different than today,
not including the ongoing maintenance and development that happens with
online-heavy games. The long-term job instability and extreme working hours
were also pervasive, much like today.</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">There
were some common recurrent threads that I never thought about: that a lot of
the small two-dimensional sprites in games of the time (like Age of Empires)
were actually pre-rendered images of much higher resolution 3D models. For some
reason, I thought there was a lot more pixel art. This general strategy of
creating high resolution images and scaling them down for the finished product
shows up in several postmortems. When Myst 3 was being made, high definition
televisions had just started being sold, and so initially the team didn't
bother to record the live action sections with HD video, which they later
regretted when trying to insert the standard definition video into the game,
which looked grainy compared to everything else.</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">Another
issue I'd never considered before was the concessions needed for localization.
In Age of Empires, the menu buttons with words (instead of icons) needed to
have a lot more space than English typically needs in order to make room for
longer translations. Other games with voice acting needed to set events to
trigger whenever certain voice clips finish, rather than at a particular time,
otherwise the localized voice clips would need to be spoken unnaturally fast or
slow to match the timing of the English version. (If you've ever been forced to
name your game character something that was 4 letters long in a NES or SNES era
game, it was probably because it was originally in Japanese, and 4 kanji would
have been plenty for most names)</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">Many of
the postmortems emphasized their choice of development tools under what went
right/wrong. Several of them mentioned 3DS Max by name as being a good choice for
modelling. Others mentioned regretting not building something for DirectX, or
being too dependent on a scripting tool they were to receive from their
publisher halfway through development. These days, Unity has become the
de-facto default for many developers, but finding, evaluating, and using the
right development engine was a much harder task back then.</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">Most of
the discussions on project planning are still relevant today. There's two cited
strategies for ordering tasks that sort-of compete. One strategy is to do as
much of the planning as possible before creating a minimum viable product. This
approach minimizes the amount of work, especially artwork, that needs to be
thrown out and redone because story elements are less likely to be dropped or
changed if it's all be planned out first. The other strategy is to get some
basic version of a playable game created as soon as possible so that people can
start testing and experimenting with features right away. The goals of this
strategy are to motivate people with a playable vision of what the game could
be, and to maximize the number of cycles of testing-and-improvement that can be
fit into the late stages of development.</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">There's a
postmortem by Peter Molyneux on Black and White, in which he talks about his
bad habit of trying to build bigger games than are feasible. Around 2011
when his mobile project "Curiosity: What's Inside the Box" started,
there were jokes that Molyneux overpromises and underdelivers, so it was
interesting to see that bit of self awareness. If you're not familiar with
Molyneux, he made quirky, experimental games. Think Bennett Foddy (Getting Over
It, Universal Paperclips), but with a larger budget.</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">A
recurring issue was scaling to meet computer requirements. Each game has a CPU
speed budget, a RAM budget, and a total file size budget which was usually a
single CD-ROM, or about 700 Mb. The Postmortem of Star Trek Voyager: Elite
Force talked about how the levels had to be either shrunk down, cut in half, or
both to accommodate all the curved surfaces that the Quake 2 allowed, but which
took much more resources than the equivalent levels in Quake 1. The Postmortem
for Age of Empires talks about the difficulty in predicting the frame rate of
the final game from prototypes in which many of assets are not yet loaded in
and running. In developing Tropico, the ability to zoom close in and see the
details of the people walking around became a major headache because it took a
disproportionate amount of disk space and RAM to handle that much detail. While
RAM limitations are less of an issue today, and disk space is almost a
non-issue, optimization for limiting computing power is still common problem.
Previously, games could rely on Moore's Law to improve computer hardware
dramatically every couple of years, but today we've had to get a lot more
creative to improve speed by distributing load across multiple cores, and by
writing non-graphical code for specialist graphical processing units.</span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span style="font-family: "Times New Roman",serif; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: EN-CA;">The book
"Postmortems" doesn't deserve to be forgotten, and a lot of the
material is relevant for game and other software development today, even though
a large section of the industry and player base has moved on to mobile.</span></div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvD8V6OeR2aPHigVB08gCYBvbJB0frrYS26zXpxnT0n6lw8Oc_tFFrh7pnxBSetPuSxpWy0I9tZqBmpudragB4QIO8b5zb6Uq_wZpm70G-BTn9YiXhnZ53BhoSfo9HR-eFPUj3SmF1x2_s/s1600/Postmortem.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="336" data-original-width="296" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvD8V6OeR2aPHigVB08gCYBvbJB0frrYS26zXpxnT0n6lw8Oc_tFFrh7pnxBSetPuSxpWy0I9tZqBmpudragB4QIO8b5zb6Uq_wZpm70G-BTn9YiXhnZ53BhoSfo9HR-eFPUj3SmF1x2_s/s320/Postmortem.PNG" width="281" /></a></div>
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Jack Davishttp://www.blogger.com/profile/16675781493059962337noreply@blogger.com0