Betting mathematics
How to calculate implied probability from odds
The implied probability of decimal odds is 1 divided by the odds: odds of 2.50 mean 1 / 2.50 = 0.40, or 40%. Across one market these probabilities add up to more than 100% because of the bookmaker's margin, so the fair probability is each one divided by that total.
Contents
Key terms
- Decimal odds
- The total returned per unit staked, stake included. Odds of 2.50 on a stake of 100 return 250.
- Implied probability
- The probability the odds assume: 1 / odds.
- Margin (overround)
- How far the implied probabilities of all outcomes add up beyond 100%. That part is what the bookmaker keeps.
- Fair odds
- Odds without the margin: 1 / margin-free probability.
The formula
Decimal odds tell you how much comes back per unit staked. In a fair bet the average return equals the stake, so p × odds = 1. That is why the probability is the reciprocal of the odds.
p = 1 / odds1 / 2.50 = 0.40 → 40%
Odds of 2.50 assume the outcome happens 40 times out of 100, before the margin is taken into account.
Odds and probabilities
The lower the odds, the more likely the outcome. Odds of 2.00 are exactly 50%.
| Odds | 1 / odds | Probability |
|---|---|---|
| 1.20 | 0.8333 | 83.33% |
| 1.50 | 0.6667 | 66.67% |
| 2.00 | 0.5000 | 50.00% |
| 2.50 | 0.4000 | 40.00% |
| 3.00 | 0.3333 | 33.33% |
| 5.00 | 0.2000 | 20.00% |
| 10.00 | 0.1000 | 10.00% |
Why the total is above 100%
The outcomes 1, X and 2 cover everything that can happen, so their probabilities should add up to exactly 100%. The bookmaker sets each price a little below the fair odds, which is why the total comes out higher. The part above 100% is the margin.
| Outcome | Odds | 1 / odds |
|---|---|---|
| 1 | 2.10 | 47.62% |
| X | 3.40 | 29.41% |
| 2 | 3.60 | 27.78% |
| Total | 104.81% |
The total is 104.81%, so this market carries a margin of 4.81%.
Probability without the margin
The simplest way to remove the margin is to divide each probability by the total. This is proportional normalisation: the ratio between outcomes stays the same and the total becomes 100%.
fair p = (1 / odds) / sum of all (1 / odds)
fair odds = 1 / fair p| Outcome | Odds | Without margin | Fair odds |
|---|---|---|---|
| 1 | 2.10 | 45.43% | 2.20 |
| X | 3.40 | 28.06% | 3.56 |
| 2 | 3.60 | 26.50% | 3.77 |
For outcome 1: 0.4762 / 1.0481 = 0.4543, so the fair odds are 1 / 0.4543 = 2.20. The rounded values add up to 99.99%; the 0.01 gap comes from rounding only.
The AI model performance page compares the model's probabilities with the market and computes the market probability exactly this way: 1 / odds, divided by the total over all outcomes of the same market. A market whose total is below 100% or above 140% is left out of the comparison.
Limitations
- Probability from odds is the market's estimate, not the true probability of the outcome.
- Proportional normalisation assumes the margin is spread evenly. Bookmakers often load more of it onto longshots, so other methods (for example the power method or Shin's method) give slightly different numbers.
- Odds move until kick-off. The same calculation with the closing odds gives a different number.
- One match proves nothing: 40% means the outcome happens 4 times in 10 on average, not that it will not happen tonight.
Related guides
Sources
- Štrumbelj, E. (2014). On determining probability forecasts from betting odds. International Journal of Forecasting, 30(4), 934–943.
- Shin, H. S. (1993). Measuring the incidence of insider trading in a market for state-contingent claims. The Economic Journal, 103(420), 1141–1153.