In probability theory and statistics the odds in favor of an event or a proposition are the quantity p / (1 − p), where p is the probability of the event or proposition. The logarithm of the odds is the logit of the probability.Odds have long been the standard way of representing probability used by bookmakers, though the method of presenting odds varies by location.

Taking an event with a 1 in 5 probability of occurring (i.e. 0.2 or 20%), then the odds are 0.2 / (1 − 0.2) = 0.2 / 0.8 = 0.25. If you bet 1 at fair odds and the event occurred, you would receive back 4 plus your original 1 stake. This would be presented in fractional odds of 4 to 1 against (written as 4 : 1 or 4/1), in decimal odds as 5.0 to include the returned stake, in craps payout as 5 for 1, and in moneyline odds as +400 representing the gain from a 100 stake.

By contrast, for an event with a 4 in 5 probability of occurring (i.e. 0.8 or 80%), then the odds are 0.8 / (1 − 0.8) = 4. If you bet 4 at fair odds and the event occurred, you would receive back 1 plus your original 4 stake. This would be presented in fractional odds of 4 to 1 on (written as 1 : 4 or 1/4), in decimal odds as 1.25 to include the returned stake, in craps as 5 for 4, and in moneyline odds as −400 representing the stake necessary to gain 100.

The odds are a ratio of probabilities; an odds ratio is a ratio of odds, that is, a ratio of ratios of probabilities. Odds-ratios are often used in analysis of clinical trials. While they have useful mathematical properties, they can produce counter-intuitive results: in the example above an 80% probability is four times the chance of a 20% probability but the odds are 16 times higher.

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