Law of large numbers

Formally, if you draw independent observations from a distribution with mean μ, the sample mean of the first n observations approaches μ as n grows without bound. The same statement in proportion form: for repeated independent trials of an event with probability p, the relative frequency of that event converges to p. Short runs can look wildly unrepresentative — ten heads in a row is unremarkable — but that noise is progressively diluted as the sample grows.

A concrete case: a fair six-sided die has expected value 3.5. Roll it twenty times and the average might be 3.9 or 3.1. Roll it twenty thousand times and the average will sit very close to 3.5. What shrinks is the average deviation, not the absolute one — the running total of heads minus tails can drift further from zero even as the proportion tightens around one half.

This is exactly where the gambler's fallacy goes wrong. The law does not say a run of losses will be balanced by compensating wins; independent trials have no memory. Convergence comes from swamping early results with a much larger volume of later ones, not from any corrective force.

Insurance rests directly on this principle. An insurer cannot predict whether any one policyholder will file a claim, but across a large pool of similar, independent exposure units it can estimate total losses with enough accuracy to set a premium that covers claims and expenses. That is why underwriters group like risks together and why a larger book of business is more predictable than a small one.

AP Statistics tests the law of large numbers in the probability and random variables unit, often by asking you to identify the gambler's fallacy or to reason about long-run relative frequency. The life and health insurance licensing exams cover it under transferring losses and underwriting, as the mathematical basis for pooling risk.