Odds calculator

How do I convert American odds to a probability?

For positive odds, 100 ÷ (odds + 100): +150 is 100 ÷ 250 = 40%. For negative odds, odds ÷ (odds + 100) using the number without the minus: −200 is 200 ÷ 300 = 66.7%.

What are decimal odds?

The total return per $1 staked, stake included: 2.50 means $2.50 back for $1. The implied probability is 1 ÷ 2.5 = 40%.

What are fractional odds?

Profit over stake, as in 3/2: win $3 for every $2 staked. The implied probability is 2 ÷ (3 + 2) = 40%.

How much do I win on a bet?

At +150, a $100 bet wins $150 profit ($250 back); at −200 you must bet $200 to win $100.

Why do the probabilities of both sides add up to more than 100%?

Bookmakers build in a margin (the "vig"), so implied probabilities are slightly inflated. The true chance of an event is not known from the odds alone. As an illustration, a market that offers +150 on one side and −200 on the other implies 40.0% + 66.7% = 106.7%; the extra 6.7 points are the margin. Scaling both down in proportion, a common approximation, gives 37.5% and 62.5%.

What is the difference between odds and probability?

Probability compares the chance of an event with all possibilities; odds compare it with the chance of the event not happening. Bland and Altman give a die example: the odds of throwing a six are 1 to 5, while the probability is 1 in 6. The conversions are odds = p ÷ (1 − p) and p = odds ÷ (1 + odds).

Odds for or odds against?

“3 to 1 against” means 3 failures for each success, a probability of 25%; “3 to 1 in favor” means 3 successes for each failure, a probability of 75%. When the format is fractional or decimal, check which side the number describes before converting.

Common mistakes with odds

First, treating odds as probability: odds of 1 to 5 mean a probability of 1/6, not 1/5. Second, reading the implied probability as the real chance. It includes the bookmaker’s margin, and it says nothing about how good the estimate is. Third, mixing formats: decimal odds include the returned stake, fractional odds do not, so 2.50 decimal equals 3/2 fractional, not 5/2. Fourth, dropping the sign on American odds, which changes the meaning: +150 and −150 are very different bets. Fifth, confusing the odds ratio of a medical study with a risk ratio; the two are close only when the outcome is rare, as the relative risk calculator shows.

One probability in every format

The same chance written four ways, from the conversion formulas above:

• 25% = +300 American = 4.00 decimal = 3/1 fractional (odds against)
• 20% = +400 American = 5.00 decimal = 4/1 fractional (odds against)
• 10% = +900 American = 10.00 decimal = 9/1 fractional (odds against)
• 5% = +1900 American = 20.00 decimal = 19/1 fractional (odds against)

A 50% chance is even money: 2.00 in decimal, 1/1 in fractional and ±100 in American form (written +100 or −100). Below 50% the American number is positive (an underdog), above 50% it is negative (a favorite). There is no American price between −100 and +100, which is why comparing two prices is easiest in decimal form or as implied probability.

Worked example: combining two outcomes

If two independent events pay 2.00 and 1.50 in decimal odds, a bet that both happen pays 2.00 × 1.50 = 3.00. The implied probabilities multiply too: 50% × 66.7% = 33.3%, which is 1 ÷ 3.00. The bookmaker's margin compounds the same way: each leg carries its own, so a combined bet costs more in margin than a single one. This works only for independent events; two outcomes in the same game are usually linked, and the product is then wrong.

Sources: Bland, J. M. and Altman, D. G. (2000), “The odds ratio”, BMJ 320:1468 (odds compared with probability). The American, decimal and fractional conversions follow the standard definitions of those formats. Related: probability calculator, relative risk calculator, Bayes’ theorem calculator, expected value calculator and lottery odds calculator.