Odds calculator

The formulas

Odds a : b in favour → p = a ÷ (a + b) Odds a : b against → p = b ÷ (a + b) Probability p → odds in favour = p : (1 − p)

Betting odds are the same idea written for payouts:

Fractional a/b → p = b ÷ (a + b) Decimal d → p = 1 ÷ d American +A → p = 100 ÷ (A + 100) American −A → p = A ÷ (A + 100)

A worked example

A bag holds 3 red balls and 2 blue ones. The odds of drawing red are 3 : 2 in favour: three ways to succeed against two ways to fail. As a probability:

p = 3 ÷ (3 + 2) = 3/5 = 0.6

The odds against red are the same numbers swapped, 2 : 3, and the probability of drawing blue is 0.4. Written as fair betting odds, 60% is fractional 2/3, decimal 1.6667 (1 ÷ 0.6) and American −150 (you would stake 150 to win 100, since 100 × 0.6 ÷ 0.4 = 150).

Switch the first box to “Fractional betting odds” to see the second default, 5/2. That price pays 5 for every 2 staked, so the implied chance of winning is 2 ÷ 7 = 0.285714, or 28.57%. The same price is decimal 3.5 and American +250.

How to read the result

Odds and probabilities carry the same information, but odds grow without limit as an event gets more likely, while probabilities stop at 1. Odds of 1 : 1 (“evens”) are a probability of 0.5. Odds in favour above 1 : 1 mean the event is more likely than not.

For betting, the probability is the break-even point: a bet at decimal odds of 3.5 makes money in the long run only if the true chance of winning is above 28.57%. Real bookmaker prices include a margin, so the fair odds shown here are longer than what you will usually be offered.

Odds in statistics

Odds also appear outside betting. In medical research and logistic regression, results are often reported as an odds ratio: the odds of an outcome in one group divided by the odds in another. If 60% of treated patients recover (odds 1.5) and 40% of untreated ones do (odds 0.667), the odds ratio is 1.5 ÷ 0.667 = 2.25. Converting between odds and probability, as this calculator does, is the first step in reading those results.

Common mistakes

• Reading odds as a fraction of the total. Odds of 3 : 2 are not a probability of 3/2 or 2/3; the total is 3 + 2 = 5.
• Mixing up “for” and “against”. Bookmakers' 4/1 is odds against, a 20% chance, not 80%.
• Forgetting the stake in decimal odds. Decimal 2.5 includes the 1 you bet; the fractional equivalent is 3/2.
• Dropping the sign on American odds. +150 and −150 imply 40% and 60%.