The Law of Large Numbers: Why More Flips Always Wins
Flip a coin 10 times and you might get 70% heads. Flip it 1,000,000 times and you will be within 0.05% of 50%. The law of large numbers guarantees this convergence.
Flip a fair coin ten times and count the heads. You might get 3 heads, or 7, or even 9. The probability of getting exactly 5 out of 10 is only about 25%. Short runs are dominated by variance. The outcome feels random because it is.
Flip the same coin 10,000 times. The fraction of heads will be extremely close to 50% — almost certainly within 1% of it. The law of large numbers guarantees this convergence. The more trials you run, the closer the empirical frequency gets to the theoretical probability.
The formal statement
Let X₁, X₂, X₃, ... be independent, identically distributed random variables with expected value μ. The weak law of large numbers states that for any ε > 0:
lim(n→∞) P(|X̄ₙ − μ| > ε) = 0 In plain language: as n grows, the probability that the sample mean deviates from the true mean by more than any fixed amount ε goes to zero.
How fast does it converge?
The standard error of the mean is σ/√n, where σ is the standard deviation of the underlying distribution. For a fair coin, σ = 0.5.
| Flips (n) | Standard error | Typical deviation from 50% |
|---|---|---|
| 100 | 0.050 | ~5% |
| 1,000 | 0.016 | ~1.6% |
| 10,000 | 0.005 | ~0.5% |
| 1,000,000 | 0.0005 | ~0.05% |
Precision improves with the square root of n. To halve your error, you need four times as many trials.
The gambler's fallacy
What the law guarantees is not that short-term deviations get corrected — it is that they get diluted. A run of 60 heads in the first 100 flips (60% heads) does not need to be followed by 40 tails to balance out. Over 10,000 total flips, those 10 extra heads are swamped by the subsequent 9,900 flips that average 50%. The proportion converges because the denominator grows, not because the numerator corrects.
Our Coin Flip Simulator lets you run up to 1,000,000 flips in the browser. Watch what happens to the heads percentage as you scale up: at 10 flips it swings wildly, at 1,000 it has mostly settled, at 100,000 it barely moves. The chi-square test quantifies how statistically normal your distribution is.