Betting mathematics
Betting variance: how far luck moves you from the average
Variance measures how far betting results move away from expected value. A bet at odds of 2.00 with an estimate of p = 0.525 has an EV of +5%, but a standard deviation of almost one whole stake. That is why after 100 such bets the result is down in 27.38% of cases, and after 1,000 in 5.32%. In 1,000 bets a streak of at least 8 losses in a row has a probability of 74.54%, even though every bet has positive EV.
Contents
Key terms
- Variance
- The average squared deviation of the result from the expected value. The larger it is, the more the result swings around the average.
- Standard deviation (σ, SD)
- The square root of the variance, in the same units as the stake. Over a long run of bets it grows with the square root of the number of bets.
- Losing streak
- Consecutive losses. They happen even when every bet has positive expected value.
- ROI
- Return on investment: the net result divided by the total staked.
The variance formula for a bet
A bet with a stake of 1 returns odds − 1 with probability p and −1 with probability 1 − p. The expected value is p × odds − 1, and the standard deviation is:
σ = odds × √(p × (1 − p))For a run of n independent bets the expected result is n × EV and the standard deviation is σ × √n. The expected result grows with n, the deviation only with √n, so an edge only shows over a long run.
Odds of 2.00 and an estimate of p = 0.525, so EV = +5% per bet. The standard deviation is 2.00 × √(0.525 × 0.475) = 0.9987, almost a whole stake.
| Bets | Expected | SD | Down | Exactly even |
|---|---|---|---|---|
| 100 | +5.00 | 9.99 | 27.38% | 7.02% |
| 500 | +25.00 | 22.33 | 12.22% | 1.91% |
| 1,000 | +50.00 | 31.58 | 5.32% | 0.72% |
The probabilities are exact, from the binomial distribution. After 100 bets with positive EV the result is down in more than one case in four.
How does variance depend on the odds?
The same EV of +5% at different odds does not mean the same risk. At higher odds a bet wins less often, so the result swings more.
| Odds | Estimate p | σ per bet | SD after 100 | Down after 100 | Down after 1,000 |
|---|---|---|---|---|---|
| 1.50 | 0.700 | 0.6874 | 6.87 | 22.07% | 1.09% |
| 2.00 | 0.525 | 0.9987 | 9.99 | 27.38% | 5.32% |
| 3.00 | 0.350 | 1.4309 | 14.31 | 38.03% | 13.68% |
| 5.00 | 0.210 | 2.0365 | 20.37 | 36.42% | 20.82% |
After 1,000 bets at odds of 5.00 the result is down in one case in five, and at odds of 1.50 in one case in a hundred. Over 100 bets the numbers do not rise smoothly because wins come in whole numbers: at odds of 5.00 a profit needs at least 21 wins, and at odds of 3.00 at least 34.
How common are losing streaks?
At odds of 2.00 and p = 0.525 a bet loses 47.5% of the time. The probability that a run of bets contains at least one streak of this many losses in a row:
| Losses in a row (at least) | In 100 bets | In 1,000 bets |
|---|---|---|
| 6 | 45.21% | 99.81% |
| 7 | 24.24% | 94.57% |
| 8 | 12.12% | 74.54% |
| 10 | 2.79% | 26.32% |
| 12 | 0.62% | 6.63% |
| 15 | 0.06% | 0.73% |
In 1,000 bets with positive EV a streak of 8 losses in a row is more likely than not. A losing streak on its own therefore does not prove the estimates are bad, just as a winning streak does not prove they are good.
What does variance mean for stake size?
A stake of 2% of the current bankroll, then 10 losses in a row: 0.98^10 = 81.71% of the bankroll is left, a drop of 18.29%. With a stake of 5% the same streak leaves 59.87%.
- The larger the stake as a share of the bankroll, the deeper the drop from the same losing streak. How stake size changes risk is compared in the Kelly criterion and flat vs Kelly guides.
- A progression after losses, as in the Martingale betting system guide, does not reduce variance, it moves it into rare, large losses.
- How many bets it takes for an edge to stand out from luck is calculated in the sample size guide.
Limitations
- The examples assume that the estimate p is correct and that bets are independent. Bets on the same match or the same day are often not independent, so the real variance is larger.
- The standard deviation describes a typical deviation. A single run can go much further than one or two SDs.
- The numbers do not include tax on winnings, the margin in the odds actually taken or changes in stake during the run.
- A short run of wins or losses says little about the quality of the estimates.