Football statistics
Sample size in betting: when a result stops being luck
A hit rate on a small sample depends heavily on luck. Over 100 bets with a 55% hit rate the 95% confidence interval runs from 45.25% to 64.75%, and over 1,000 bets from 51.92% to 58.08%. With no edge at all, 60 or more wins out of 100 at odds of 1.80 happens in 21.41% of cases. For an edge of +5% at odds of 2.00 to stand out from luck, about 1,533 bets are needed.
Contents
Key terms
- Sample
- The set of bets, matches or predictions on which something is measured. The sample size is how many there are.
- Hit rate
- The share of bets that won. It says nothing without the odds: 55% is excellent at odds of 2.00 and poor at odds of 1.50.
- Confidence interval (95%)
- The range that would contain the true value in 95% of repeated samples. The larger the sample, the narrower the interval.
- Standard error
- How much the hit rate moves from sample to sample on average: √(p × (1 − p) / n).
How much does a hit rate move by luck?
± 1.96 × √(p × (1 − p) / n)| Bets | ± percentage points | 95% interval |
|---|---|---|
| 20 | ±21.80 | 33.20% – 76.80% |
| 50 | ±13.79 | 41.21% – 68.79% |
| 100 | ±9.75 | 45.25% – 64.75% |
| 250 | ±6.17 | 48.83% – 61.17% |
| 500 | ±4.36 | 50.64% – 59.36% |
| 1,000 | ±3.08 | 51.92% – 58.08% |
| 5,000 | ±1.38 | 53.62% – 56.38% |
The interval narrows with the square root of the number of bets: to halve its width you need a sample four times larger.
Example: 60 wins out of 100
Someone has 60 wins from 100 bets, all at odds of 1.80. The ROI is 0.60 × 1.80 − 1 = +8%.
The break-even rate is 1 / 1.80 = 55.56%. If the true probability of every bet is exactly that, so with no edge at all:
- 60 or more wins out of 100 happens in 21.41% of cases;
- 600 or more wins out of 1,000 happens in 0.25% of cases.
The same hit rate on a sample ten times larger means something completely different. Over 100 bets +8% easily comes from luck, over 1,000 rarely.
For small samples and rates close to 0% or 100% the normal approximation is poor, so the Wilson interval is used. For 60 out of 100 it runs from 50.20% to 69.06% and includes the break-even rate of 55.56%. For 600 out of 1,000 it runs from 56.93% to 62.99%, entirely above it.
How many bets does it take to show an edge?
The number of bets after which the expected profit is 1.96 standard deviations above zero:
n ≈ 1.96² × σ² / EV²
σ² = (1 + EV) × (odds − 1 − EV)| Odds | EV +5% | EV +2% |
|---|---|---|
| 1.50 | 727 | 4,703 |
| 2.00 | 1,533 | 9,601 |
| 3.00 | 3,147 | 19,397 |
| 5.00 | 6,374 | 38,989 |
A smaller edge needs far more bets: an EV of +2% instead of +5% needs a sample more than six times larger. Higher odds need more bets, because each bet swings more. Why is shown in the betting variance guide.
Trends and small samples in football
Football statistics often rest on small samples. A league of 20 clubs has 380 matches per season, and each team plays 19 at home.
- “The team scored in 8 of its last 10 matches”: 80%, but the 95% Wilson interval runs from 49.02% to 94.33%.
- “The team scored in 15 of its 19 home matches”: 78.95%, and the interval runs from 56.67% to 91.49%.
Both trends are true, but they do not show that the chance of a goal next time is exactly 80%. A trend is a fact about past matches; an estimate for the next match needs other data too.
How StatsAI handles sample size on the AI model performance page:
- the interval for the difference in Brier score between the AI model and the market comes from resampling whole days, 1,000 repetitions, because matches on the same day are not independent;
- calibration shows the 95% Wilson interval, and bins with fewer than 100 predictions are not drawn;
- league groups are only shown when they have at least 1,000 events.
Limitations
- The formulas assume independent bets with the same probability. Bets on the same match, day or league are often not independent, so a larger sample is needed.
- The number of bets in the table is approximate and does not guarantee that the result will not be down.
- A hit rate without odds says nothing about profitability. Compare it with the break-even rate of 1 / odds.
- Past results on a large sample still do not mean the same result in the future, because markets and teams change.
Related guides
Sources
- Wilson, E. B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22(158), 209–212.
- Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of Statistics, 7(1), 1–26.