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Odds

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 0.2/0.8=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 by a British bookmaker as odds of 4 to 1 against (written as 4/1), by a European bookmaker as 5.0 to include the returned stake, and by an American bookmaker 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 0.8/0.2=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 by a British bookmaker as odds of 4 to 1 on (written as 1/4), by a European bookmaker as 1.25 to include the returned stake, and by an American bookmaker as −400 representing the stake necessary to gain 100.

The odds are a ratio of probabilities; an odds-ratio is a ratio of odds, i.e., a ratio of ratios of probabilities. Odds-ratios are often used in analysis of clinical trials.
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