In probability theory, odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics.

Odds also have a simple relation with probability: the odds of an outcome are the ratio of the probability that the outcome occurs to the probability that the outcome does not occur. In mathematical terms, where p is the probability of the outcome:

${\displaystyle {\text{odds}}={\frac {p}{1-p}}}$

where 1 – p is the probability that the outcome does not occur.

Odds can be demonstrated by examining rolling a six-sided die. The odds of rolling a 6 is 1 to 5 (abbreviated 1:5). This is because there is 1 event (rolling a 6) that produces the specified outcome of "rolling a 6", and 5 events that do not (rolling a 1, 2, 3, 4 or 5). The odds of rolling either a 5 or 6 is 2:4. This is because there are 2 events (rolling a 5 or 6) that produce the specified outcome of "rolling either a 5 or 6", and 4 events that do not (rolling a 1, 2, 3 or 4). The odds of not rolling a 5 or 6 is the inverse 4:2. This is because there are 4 events that produce the specified outcome of "not rolling a 5 or 6" (rolling a 1, 2, 3 or 4) and two that do not (rolling a 5 or 6).

The probability of an event is different, but related, and can be calculated from the odds, and vice versa. The probability of rolling a 5 or 6 is the fraction of the number of events over total events or 2/(2+4), which is 1/3, 0.33 or 33%.

When gambling, odds are often the ratio of winnings to the stake and you also get your wager returned. So wagering 1 at 1:5 pays out 6 (5 + 1). If you make 6 wagers of 1, and win once and lose 5 times, you will be paid 6 and finish square. Wagering 1 at 1:1 (Evens) pays out 2 (1 + 1) and wagering 1 at 1:2 pays out 3 (1 + 2). These examples may be displayed in different forms, explained later:

• Fractional odds with a slash: 5 (5/1 against), 1/1 (Evens), 1/2 (on) (short priced horse). Fractional odds can also be written with a colon or a hyphen or dash.
• Tote boards use decimal or Continental odds (the ratio of total paid out to stake), e.g. 6.0, 2.0, 1.5
• In the US Moneyline a positive number lists winnings per $100 wager; a negative number the amount to wager in order to win $100 on a short-priced horse: 500, 100/–100, –200.