Betting mathematics
How to remove the margin from odds (de-vig)
The margin is removed by scaling the implied probabilities so that they add up to 100%. The most common way is the proportional method: each 1 / odds is divided by the total. The additive and power methods spread the margin differently, so they give a longshot higher fair odds.
Contents
Key terms
- De-vig
- Removing the bookmaker margin from odds to get probabilities that add up to 100%.
- Fair probability
- The estimated probability of an outcome once the margin is removed.
- Fair odds
- 1 / fair probability.
- Favourite-longshot bias
- The tendency for longshot odds to be shorter than fair on average, so the margin on them is larger.
Proportional method
Each implied probability is divided by the total of all implied probabilities in the same market. The ratios between outcomes stay the same and the total becomes 100%.
fair p = (1 / odds) / Σ (1 / odds)Odds of 2.10 / 3.40 / 3.60, total 104.81%.
| Outcome | 1 / odds | Fair p | Fair odds |
|---|---|---|---|
| 1 | 47.62% | 45.43% | 2.20 |
| X | 29.41% | 28.06% | 3.56 |
| 2 | 27.78% | 26.50% | 3.77 |
Additive method
The margin is split into equal parts and the same part is subtracted from every outcome. In a three-way market with a 4.81% margin that is 1.60 percentage points per outcome.
fair p = (1 / odds) − (Σ (1 / odds) − 1) / number of outcomes| Outcome | 1 / odds | Fair p | Fair odds |
|---|---|---|---|
| 1 | 47.62% | 46.02% | 2.17 |
| X | 29.41% | 27.81% | 3.60 |
| 2 | 27.78% | 26.17% | 3.82 |
Because the same amount comes off every outcome, the favourite loses the least in relative terms and the higher-odds outcomes lose the most.
Power method
The method looks for a number k above 1 such that the probabilities raised to the power k add up to exactly 100%. Raising to a power shrinks a small probability relatively more than a large one, so this method takes more margin off the longshot.
fair p = (1 / odds)^k, where Σ (1 / odds)^k = 1For odds of 2.10 / 3.40 / 3.60, k = 1.0461.
| Outcome | 1 / odds | Fair p | Fair odds |
|---|---|---|---|
| 1 | 47.62% | 46.02% | 2.17 |
| X | 29.41% | 27.80% | 3.60 |
| 2 | 27.78% | 26.18% | 3.82 |
In this market the additive and power methods give almost the same numbers, while the proportional method puts the favourite about 0.6 percentage points lower.
When the methods disagree
The differences grow when a market has a strong favourite and a big longshot.
Odds of 1.30 / 5.50 / 11.00, total 104.20%. The table shows fair odds.
| Method | 1 | X | 2 |
|---|---|---|---|
| Proportional | 1.35 | 5.73 | 11.46 |
| Additive | 1.32 | 5.96 | 13.00 |
| Power | 1.32 | 6.09 | 12.71 |
For the longshot, the proportional method gives a fair probability of 8.72% and the power method 7.87%. If the true probability is closer to the second number, odds of 11.00 on the longshot look better than they are when only the proportional method is used.
Which method is closer to reality depends on how the bookmaker spreads its margin. On a large sample of odds, Štrumbelj (2014) found that Shin's method, which starts from the favourite-longshot bias, gives more accurate probabilities than the proportional one. Shin's method has to be solved iteratively, so it is not worked through with numbers here.
What StatsAI uses
The AI model performance page computes the market probability with the proportional method: 1 / odds, divided by the total of all outcomes in the same market. The same method is used in Implied probability and CLV, so the numbers in those guides can be compared.
Limitations
- No method reveals the true probability. Each one is an assumption about how the bookmaker spread its margin.
- The additive method can give a negative probability when 1 / odds of an outcome is below margin / number of outcomes. With a 110% total and three outcomes it subtracts 3.33 points, while odds of 41.00 carry only 2.44%.
- In a two-way market with similar odds the methods differ very little. The gap grows with the margin and with the distance between the odds.
- Fair odds are the market's estimate at one moment. The same calculation with closing odds gives a different number.
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.
- Clarke, S., Kovalchik, S., Ingram, M. (2017). Adjusting bookmaker's odds to allow for overround. American Journal of Sports Science, 5(6), 45–49.