Odds

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In probability theory, odds provide a measure of the likelihood of a particular outcome. When specific events are equally likely, odds 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 have a simple relationship with probability. When probability is expressed as a number between 0 and 1, the relationships between probability p and odds t are:

${\displaystyle t={\frac {p}{1-p}}}$

${\displaystyle p={\frac {t}{1+t}}}$

When probability is expressed as a percentage, it must be divided by 100 before using these formulas. When the odds have value t, one often says "t to 1" or writes "t:1". If the value t can be written as a fraction p / q then one can say "p to q" or write "p:q".

Another way to express odds is using "for" instead of "to": "f for 1" or "r for q" where

${\displaystyle f=t+1}$

${\displaystyle r=p+q}$

Odds can be demonstrated by examining rolling a six-sided die. The odds of rolling a 6 is "1 to 5" or "1:5". This is because there is 1 event (rolling a 6) that produces the specified outcome, 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, 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 or approximately 0.33 or 33%.^[1]

When gambling, odds are often given as the ratio of the possible net profit to the possible net loss. Typically you pay the possible loss ("stake" or "wager") up front and, if you win, you are paid the net win plus you also get your stake returned. So wagering 1 at 5:1, which is called "5 to 1", pays out 5 + 1 = 6, which is called "6 for 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, "1 to 1") pays out 1 + 1 = 2 ("2 for 1") and wagering 2 at 1:2 ("1 to 2") pays out 1 + 2 = 3 ("3 for 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 net winnings for a $100 wager; a negative number is the amount to wager in order to win $100 net. Examples: +500 ("5 to 1", "6 for 1"), +100/−100 ("Evens", "1 to 1", "2 for 1"), −200 ("1 to 2", "3 for 2").