The law of large numbers says the average result of many independent bets settles towards its expected value as the number of bets grows. A 40% chance shows up as close to 40% winners over thousands of bets, never reliably over ten.
It is why the betting strategy hub judges a method over many bets rather than a good week.
How the average settles: 40% bets, worked
Example: Illustrative: every bet has a true 40% chance and is priced at $2.60, so each $1 bet returns $2.60 or nothing.
| Bets | Expected winners | Typical gap in winners (one standard deviation) | Winners in at least 19 runs out of 20 |
|---|---|---|---|
| 10 | 4 | 1.55 | 1 to 7 (10% to 70%) |
| 100 | 40 | 4.90 | 31 to 50 (31% to 50%) |
| 1,000 | 400 | 15.49 | 370 to 430 (37% to 43%) |
| 10,000 | 4,000 | 48.99 | 3,904 to 4,096 (39.0% to 41.0%) |
One standard deviation is the square root of (bets x 0.4 x 0.6). The last column is the narrowest range of winners with at least a 95% chance (the likeliest where several tie), from the exact binomial distribution. Down the table the gap in winners grows, yet as a share of the bets it falls from 1.55 / 10 = 15.5 points to 48.99 / 10,000 = 0.49 points. The count wanders further from expected while the average closes in on 40%.
Why the law of averages is wrong: dilution, not repayment
Say those 40% bets start with 1 winner from 10, three short of the expected 4. The next 990 bets expect 0.4 x 990 = 396 winners, so after 1,000 bets the expected total is 1 + 396 = 397: still three short. The win rate has recovered from 10% to an expected 39.7% only because the shortfall is now spread over 1,000 bets instead of 10.
Nothing in the next race or price remembers the bad start, and the popular "law of averages", the idea that results must even out soon, is the gambler's fallacy under another name.
Expected value and the long run
Expected value (EV) per $1 = chance x decimal odds - 1, so the bets above have EV = 0.40 x 2.60 - 1 = +0.04, a 4% edge. How fast that average becomes a result:
| Bets of $10 | Expected profit | Typical swing either way | Chance of being ahead |
|---|---|---|---|
| 100 | $40 | $127 | About 62% |
| 1,000 | $400 | $403 | About 84% |
| 10,000 | $4,000 | $1,274 | About 99.9% |
Typical swing = $10 x 2.60 x the square root of (bets x 0.4 x 0.6). Ahead means at least 39 winners in 100 bets (39 x $26 = $1,014 back on $1,000 staked), 385 in 1,000 and 3,847 in 10,000, and each chance is the exact binomial probability of reaching that count. Profit grows with the bets and the swing only with their square root, so the edge pulls clear in the thousands. Value betting covers finding edges, and sample size in betting how many bets a record needs to prove one.
Without an edge of your own, the same law works against you: at an illustrative margin of 5 cents in every dollar staked, losses settle near 5 cents per dollar over enough bets, the logic behind every bookmaker's margin.
Risk: Betting involves risk. A real edge can still be behind after hundreds of bets, the chance behind any EV figure is an estimate, and there is no guarantee of profit. See responsible gambling for limits and support.
What the law of large numbers does not say
- It does not say when: no number of bets forces an average onto its expected value.
- It does not fix a wrong estimate. If your 40% is really 36%, results settle on 36%, and the edge becomes 0.36 x 2.60 - 1 = -6.4%.
- Long prices settle more slowly, and bets on the same race or game rise and fall together, which slows it further.