What does it say?
Flip a coin 10 times and you might get 7 heads. Flip it 10,000 times and the share of heads will be very close to 0.5. As trials pile up, the sample average converges to the expected value. That's different from the central limit theorem, which describes how the distribution of sample averages approaches the normal distribution.
How is it different from the gambler's fallacy?
Believing black is "due" after five reds in a row is the gambler's fallacy. The law of large numbers doesn't make up for past results; new trials simply dilute early streaks. Tversky and Kahneman coined "the law of small numbers" for our tendency to expect small samples to behave like big ones. Insurers and casinos profit because, across huge numbers of policies or bets, averages become very predictable.
FAQ
- How large is "large"?
- There's no fixed number; error shrinks roughly in proportion to 1/√n.
- Does it apply to investing?
- It helps explain diversification, but returns aren't truly independent and identically distributed.
Sources reviewed: 2026-10-25
References
- Bernoulli, J. (1713). Ars Conjectandi.
- Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers. Psychological Bulletin, 76(2), 105–110.