Sometimes I hear people ask, ‘what are the odds?’ and without skipping a breath they start mentioning probabilities rather than odds.

Unless you have been taught otherwise, I guess it’s natural to think of these as vaguely the same sort of thing (not to mention similar concepts such as chance and likelihood). However, within statistics they have precise and different meanings.

You can think of probability and odds as being like Celsius and Fahrenheit. They measure the same thing but on different scales. You need to convert from one to the other before you can compare them.

A probability is a number between 0 and 1 that we use to represent how likely something is to happen. I think most people know this and use it correctly. The letter \(p\) is often used to denote a probability.

Odds are another way to give a number to an event to represent its chance of occurring, but this time the scale goes between 0 and infinity. A probability of \(p\) corresponds to an odds of \(p/(1-p)\). For example, suppose there are only two possibilities for the weather tomorrow: rainy or sunny. If the probability of rain is 0.2, then the corresponding odds of rain is 0.2/0.8 = 0.25. The probability of being sunny is therefore 0.8, and the odds is 0.8/0.2 = 4.

This version of odds is also called odds in favour. An alternative is odds against, which is simply the reciprocal, \( (1-p)/p\). In the weather example, the odds against rain is 4, and the odds against sun is 0.25.

Gambling odds

The place where most people encounter odds is at the horse races (or other sports where gambling is popular). In this case, the ‘odds’ are a way for bookmakers to show how much money they would pay if you were to select the winning bet. The favourite horse in a race will pay less than any of the others, because it is deemed most likely to win.

It turns out many different types of odds are used. (I don’t know why there are so many, it only makes things difficult!) They typically vary by country. I’ll describe three of them.

In the UK, the standard is fractional odds. It is the ratio of the winnings to the bet amount, expressed as a fraction. For example, if were offered odds of 3/2 for the horse Prancing Diva, and placed a $10 bet, you would be paid 3/2 \(\times\) $10 = $15 if it won, plus also your original $10 bet, leading to a total payout of $25. If the horse loses, you lose your $10 bet.

In Australia, the convention is to use decimal odds. This is the ratio of the full payout (winnings plus original bet) to the bet amount, expressed as a decimal. For Prancing Diva, the equivalent in decimal odds is 2.5 (which is $25 divided by $10). It is easy to calculate decimal odds from fractional odds, simply convert to a decimal and add 1 (for the original bet).

In the USA, the system of choice is moneyline odds. It is represented as a whole number. When positive, it is the winnings for a bet of 100. When negative, it is the bet required to win 100. The odds for Prancing Diva in this case would be 150. To get moneyline odds from fractional odds, simply multiply by 100 if greater than 1, and multiply the reciprocal by -100 if less than 1. For example, 3/1 becomes 300 and 1/3 becomes -300.

The table below compares these three types of odds.

Gambling odds vs true odds

Strictly speaking, gambling odds are different to the odds I described earlier, which I’ll refer to as true odds. Whereas true odds are precise statements about how likely an event is, gambling odds describe possible financial transactions on offer. You can think of them as showing the ‘cost’ of various bets.

To understand the relationship between the two, suppose there will be a race between Prancing Diva, Gallop-a-lot, Canterberry and Trotskyite, with the the probability of each horse winning being 0.5, 0.2, 0.2 and 0.1 respectively. The corresponding true odds against are 1, 4, 4 and 9.

Honest Joe the bookmaker could offer fractional odds of 1/1, 4/1, 4/1 and 9/1 on this race (i.e. equal to the true odds against). If he were to do this, his average profit would be exactly zero. This is not a good way to run a business. Instead, he offers the less favourable odds of 2/3, 3/1, 3/1 and 7/1. This reduces his required payouts and increases his profit.

If we pretend these are the true odds against and convert them back to probabilities, we get 0.6, 0.25, 0.25 and 0.125 respectively. We call these the implied probabilities.

Probabilities have the nice property that if you add them all up, you always get 1. In Joe’s case, the total is 1.225. The difference is due to him building in a profit margin. The bigger the difference, the greater the profit.

An intuitive explanation is that Joe is ‘pretending’ each horse is more likely to win then they actually are, so that he can pay you less on your bets. This requires him to add in extra probability, pushing the total over the true total probability of 1.

While gambling odds need to reflect the underlying knowledge of how likely each of the possible events are (otherwise punters would have a sure bet), the fact that they don’t respect the probability sum property means they are not true odds.

Everyday odds

When your friend asks you, ‘what are the odds?’ and starts quoting numbers, which type of odds are they (if any at all)? Most likely it’s unclear. To avoid confusion, I like to stick with probabilities. What do you prefer?