Odds ⇄ Probability Converter (fractional, decimal, American)

Understanding the odds and probability converter — concept and principle

Odds and probability describe the same uncertainty in different formats. Probability is the share of all outcomes, odds compare outcomes for and against, and bookmakers quote prices as fractional (UK), decimal (Europe, Australia) or American (moneyline) odds. This converter turns a probability in percent into all of these formats, so you can compare a quoted price with your own estimate or read statistics written as odds.

How it is calculated

Let p be the probability as a fraction (p = percent ÷ 100).

• Odds for: p : (1 − p), shown as 1 : (1 − p)/p.
• Fractional odds (against): (1 − p)/p to 1, the profit per 1 unit staked if the event happens.
• Decimal odds: 1/p, the total return per 1 unit staked, including the stake.
• American odds: if p < 50%, +100·(1 − p)/p (profit on a 100 stake); if p ≥ 50%, −100·p/(1 − p) (stake needed to win 100). The calculator rounds American odds to the nearest whole number.

Going back: p = 1/decimal; p = 100/(A + 100) for +A; p = A/(A + 100) for −A.

The principle

Odds count outcomes against each other instead of against the total. If an event happens 1 time in 4, it happens 1 time for every 3 times it does not, so the odds for are 1 : 3 and the odds against are 3 : 1. A fair price is one where the expected profit is zero: with decimal odds d, you win d − 1 with probability p and lose 1 with probability 1 − p, and p(d − 1) − (1 − p) = 0 gives d = 1/p.

Worked example

1. Default: probability 25%, so p = 0.25.
2. Odds for: (1 − 0.25)/0.25 = 3, giving 1 : 3.
3. Fractional odds against: 3/1, so a winning 1-unit bet earns 3 units of profit.
4. Decimal odds: 1/0.25 = 4, a total return of 4 per unit staked.
5. American: p < 50%, so +100 × 3 = +300.

How to read the result

Higher decimal or positive American odds mean a less likely event. Even money is 50%: 1/1, decimal 2, American +100 (shown as −100 at exactly 50%). The numbers here are fair odds with no margin. Bookmakers shorten prices so that the implied probabilities of all outcomes add up to more than 100%; the excess is their margin (overround). To compare a quote with your estimate, convert the quote to an implied probability.

Common mistakes

• Reading odds as probability: 1 : 3 means 25%, not 33%.
• Forgetting that decimal odds include the stake and fractional odds do not (decimal = fractional + 1).
• Using the sign of American odds the wrong way round: + means the underdog, − the favourite.

Frequently asked questions

How do I convert decimal odds 2.5 to probability?

1 ÷ 2.5 = 0.4, so 40% implied probability (before removing any margin).

What does −150 mean in American odds?

You must stake 150 to win 100 profit; the implied probability is 150 ÷ 250 = 60%.

Are "odds" in medical studies the same thing?

The definition is the same, p/(1 − p), and an odds ratio compares the odds of two groups. Odds and risk are close only when the event is rare.

Sources: Sheldon M. Ross, A First Course in Probability, Pearson — expected value · Alan Agresti, Categorical Data Analysis, Wiley — odds and odds ratios