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.

**Probability
and (Statistical) Odds**

Everything comes from probability, so let's define that first.

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.

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*.

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).

That means the probability of any cup being a winner was 45,000,000 / 270,000,000,

which simplifies to 45/270

and again to 1/6.

Next, let's define odds in a statistical context

The odds of an event is the fraction Pr(Event) / Pr(Not Event),

or, equivalently,

Odds = Pr(Event) / (1 - Pr(Event).

In both the coffee cup and dice examples, Pr(event) = 1/6

That means the odds of the event (e.g., rolling a 3, choosing a winning coffee cup) would be

(1/6) / (1 - 1/6) =

(1/6) / (5/6) =

(1/5)

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.

The way to say the odds is of this is "1 to 5" or "5 to 1 against". [Foot 1]

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.

For example:

"Betting on certain number in roulette pays 35-to-1" means that you'll win 35 times your bet (and your original bet).

"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).

"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.

"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).

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]

This isn't the only way the term 'odds' is used.

**Decimal
odds**

"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.

Decimal odds can be calculated as:

(Possible total payout) / (Amount bet)

where the payout includes your original bet.

For example: The decimal odds being offered on the Vancouver
Canucks to beat the Calgary Flames win are currently **2.5**.

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.

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).

From their perspective, the decimal odds of the Calgary Flames winning are:

(Possible payout) / (Amount bet) =

2.50 / 1.50 = 1.67

Which means your friend is getting a possible payout of **$1.67
for every $1 they put at risk**.

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?

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: **implied probability**.

**Implied
probability**

"Implied probability" is the probability an event has to have for a bet to neither winning nor lose money in the long run.

Implied probability can be calculated as 1 / Decimal Odds

For example, a decimal odds offer of 2.50 for the Vancouver Canucks implies that the Canucks have a 1 / 2.50 = 0.40 or 40% probability to win.

We can check our work by calculating the expected payout.

Expected payout =

Pr(Payout) * (Value of payout)

= 0.40 * 2.5 = 1

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.

E[Profit] = E[Payoff] - E[Cost] = 1 - 1 = 0

Your friend who bet against you got decimal odds of 1.67 for betting on the Flames. The implied probability of such odds are:

1 / Decimal Odds = 1 / 1.67 = 0.60 or 60%.

Your friend's expected payout is: Pr(Payout) X (Value of payout) = 0.60 * 1.67 = 1

That should make sense because then Pr(Canucks win) + Pr(Flames win) = 0.40 + 0.60 = 1. 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.)

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.

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.

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.

**American
odds**

"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.

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.

**American odds describe the net profit.**

Here, the American odds being offered on a Canucks win is **+150**.

If the American odds are positive, then they show "how much I will I profit for betting $1?".

Betting on something with +150 American odds means that a $1.00 bet will produce a $1.50 gain if the Canucks win.

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.

The American odds being offered on a Flames win is **-150**.

If the American odds are negative, then they show "how much must I bet to gain $1?".

It's a confusing system at first, but unlike other American measurement systems, this one has some logical value.

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.

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.

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.

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.

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.

**Translating
between odds**

Let A be the American odds, D be the decimal odds, and P be the implied probability.

If American odds are positive, they describe what you get for $1, not including your $1 back, so

If A > 0:

D = (A / 100) + 1

A = (D * 100) - 100

P = 100 / (100 + A)

If American odds are negative, they describe what it costs to gain $1, so the reciprocal value is used.

If A < 0:

D = 1 - 100/A

A = 100 / (1-D)

P = 1 - (100 / (100 - A))

The R code for all the American-Decimal-Implied Probability translations is here:

# Note: Enter in ONE of A, D, or P (American, Decimal, implied Probability)

# If you enter in two or more, and there are contractions

# Decimal odds precede implied probability precede American odds

convert_odds = function(A=NA, D=NA, P=NA)

{

if(!is.na(D)){P = 1/D}

if(!is.na(P)){D = 1/P}

if(!is.na(D) & D >= 2)

{

A = (D * 100) - 100

}

if(!is.na(D) & D < 2)

{

A = 100 / (1-D)

}

if(!is.na(A) & A > 0)

{

D = (A / 100) + 1

P = 100 / (100 + A)

}

if(!is.na(A) & A < 0)

{

D = 1 - 100/A

P = 1 - (100 / (100 - A))

}

return(list(A=A, D=D, P=P))

}

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.

Follow-up questions:

**Q1:** Check that all the formulas work by

A) Finding A and P when D = 2.5

B) Finding D and P when A = -150

C) Finding A and D when P = 0.50

**Q2:** 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?)

**Q3:** 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.

A) Find the decimal odds and implied probability of each side.

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?

**Q4:** Make and test your own R functions to translate
from American odds to decimal odds to implied probability.

[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.

[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.

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.

Gambling can be an addiction. Know your limit, or there's end to the mess you'll find yourself in. |

To be covered in later notes on odds:

Win / Loss / Push (and other multi-way outcomes)

Parleys/Multipliers

Asian half-odds

Prediction market (classic and Twitch)

The mechanics of vig (and other places it shows up, like poker tournaments and perimutual markets)

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