Odds converter and implied probability

This page explains the arithmetic of odds. It does not recommend bets, predict results or link to any betting service. Gambling is age-restricted and regulated differently from place to place. If gambling is causing problems for you or someone close to you, free confidential help is available, for example from 1-800-GAMBLER in the US or the National Gambling Helpline on 0808 8020 133 in the UK.

How to use it

Choose the format you have, type the price, and every other format appears below along with the implied probability and what a stake of 100 (in any currency) would return, including the stake. American odds can be typed with or without the plus sign; fractional odds as 5/2 or the word evens; a probability as a percentage like 40.

The conversions

Everything goes through decimal odds, which is the total return per unit staked:

• American +A → decimal 1 + A ÷ 100; American −A → decimal 1 + 100 ÷ A
• Fractional a/b → decimal 1 + a ÷ b
• Decimal d → implied probability 1 ÷ d
• Decimal d ≥ 2 → American +(d − 1) × 100; d < 2 → American −100 ÷ (d − 1)

Fractional odds are shown as the simplest fraction with a denominator up to 40 that matches the decimal price exactly. When none does, the nearest one is shown with a ≈ sign: the decimal price 1.91 has no tidy fraction and is shown as ≈ 10/11.

Worked example

Take American odds of −150. Decimal is 1 + 100 ÷ 150 = 1.667, fractional is 100/150 = 2/3, and the implied probability is 1 ÷ 1.667 = 60%. A stake of 100 returns 166.67, of which 66.67 is profit. The mirror price, +150, is decimal 2.50, fractional 3/2 and implies 40%.

Now the margin. When both sides of a game are priced at −110, each implies 110 ÷ 210 = 52.4%. Together they add up to 104.8%, though the true probabilities of two outcomes of one game must add to 100%. The 4.8% excess, the overround, is how the price-setter builds in a margin. Dividing each implied probability by the total, 104.8%, removes it and gives 50% each, a fair decimal price of 2.00, or +100.

Common prices in every format

What implied probability is not

An implied probability is the probability a price would represent if it were fair. It is not a forecast. Prices move with the money placed on each side, include a margin, and are rounded to conventional steps, so the implied figure is at best an approximate reflection of a market's view.

Removing the margin is also a modelling choice. This page uses the simplest method, dividing each implied probability by their total, which assumes the margin is spread evenly across outcomes. In practice margins are often heavier on long shots (the "favourite-longshot bias"), and methods such as Shin's model or the power method remove more margin from the long shot and less from the favourite. They give noticeably different "fair" probabilities for unlikely outcomes, so treat the fair column as one estimate rather than the answer.

Probability here means the same thing it does anywhere else in sport: the standings and magic number pages describe what has already happened or what must happen; odds describe what a price says might happen. The magic number calculator is the tool for "what has to happen for my team to clinch".