Betting mathematics
What is the bookmaker margin and how is it calculated
The margin is the part of the stake the bookmaker keeps through its odds. It comes from the sum of the probabilities of all outcomes in one market: the sum of 1 / odds minus 100%. Odds of 2.10 / 3.40 / 3.60 add up to 104.81%, so the margin is 4.81% and the market pays back 95.41% of stakes on average.
Contents
Key terms
- Margin (overround)
- How far the implied probabilities of all outcomes in one market add up above 100%.
- Market payout
- How much of the stake comes back on average if every outcome were backed in proportion: 1 / total.
- Fair odds
- Odds without a margin, where the average return would be exactly 100%.
The formula
Odds of 2.10 mean a probability of 1 / 2.10 = 47.62%. Done for every outcome of one market, fair odds would add up to exactly 100%. Everything above that is the margin.
margin = Σ (1 / odds) − 1
payout = 1 / Σ (1 / odds)| Outcome | Odds | 1 / odds |
|---|---|---|
| 1 | 2.10 | 47.62% |
| X | 3.40 | 29.41% |
| 2 | 3.60 | 27.78% |
| Total | 104.81% |
The margin is 104.81% − 100% = 4.81%. The payout is 1 / 1.0481 = 0.9541, or 95.41%: for every 100 staked on this market, the odds pay back 95.41 on average.
Two definitions of margin
The same idea is written down in two ways, and the two are easy to mix up.
| Definition | Calculation | Example |
|---|---|---|
| Overround | total − 1 | 4.81% |
| Share of stakes the bookmaker keeps | 1 − payout | 4.59% |
The first says how far the probabilities are inflated, the second how much of the stake does not come back on average. One person quoting 4.81% and another quoting 4.59% can be describing the same market.
Two-way markets
Markets with two outcomes (both teams to score, over/under goals) work the same way.
| Odds | Total | Margin | Payout |
|---|---|---|---|
| 1.90 / 1.90 | 105.26% | 5.26% | 95.00% |
| 1.85 / 1.95 | 105.34% | 5.34% | 94.93% |
At 1.90 / 1.90 both outcomes have the same fair odds of 2.00. At 1.85 / 1.95, 1 / 1.85 is a probability of 54.05% and 1 / 1.95 is 51.28%.
Margins on accumulators
When selections are combined on one ticket, the odds multiply, and so do the payouts. A double with a 95% payout on each leg returns 0.95 × 0.95 = 90.25% of the stake on average.
Every leg has a 95% payout (like odds of 1.90 / 1.90), the fair probability is proportional and the legs are independent.
| Legs on the ticket | Average payout | Average result |
|---|---|---|
| 1 | 95.00% | −5.00% |
| 2 | 90.25% | −9.75% |
| 3 | 85.74% | −14.26% |
| 5 | 77.38% | −22.62% |
| 10 | 59.87% | −40.13% |
That is why a ticket with many legs has high odds and also a large average loss. What changes when the same legs are played as a system is shown in Accumulator or system bet.
The margin on StatsAI
The AI model performance page compares the model with the margin-free market: each implied probability is divided by the total of all outcomes in the same market. A market whose total is below 100% or above 140% is left out, because such a total usually means an error in the odds. How the margin is removed, and why there is more than one method, is covered in Removing the margin.
Limitations
- The margin is an average for the whole market. It is usually not spread evenly across outcomes, so the margin-free probability of one outcome is only an estimate.
- The same bookmaker uses different margins on different markets and matches, and odds keep moving until kick-off.
- A 95% payout is a long-run average that assumes the margin-free odds are right. It says nothing about a single ticket.
- If your probability estimate is better than the market's, the maths changes, but only a large number of bets can show that (see expected value).