the law of large numbers
/ LLN /
The law of large numbers is the quiet promise at the heart of all insurance: as you average over more and more independent cases, the average settles down and stops jumping around, drawing ever closer to the true expected value. Toss a fair coin ten times and you might get 7 heads; toss it ten thousand times and the fraction of heads will hug 50% remarkably tightly. The chaos of individual outcomes dissolves into a stable, predictable average.
Stated carefully, if you have many independent observations from the same distribution with mean μ, their running average converges to μ as the number of observations grows without bound. The individual cases stay just as unpredictable — the next coin toss is still 50-50 — but the average of many of them becomes nearly certain. Crucially, the standard deviation of the average shrinks in proportion to one over the square root of the number of cases, so to halve the wobble in the average you need four times as many cases.
This is the single principle that makes insurance possible. No one can predict whether a particular house will burn this year, but an insurer covering a hundred thousand independent homes can predict the total claims with great confidence, and can therefore charge a premium close to the expected loss plus a modest margin. Risk pooling, the actuarial control cycle, and the whole logic of charging an average price for an uncertain individual risk all rest on the law of large numbers. Its silent condition — independence — is also its fault line: when losses are correlated (a hurricane, a pandemic), the law gives much weaker comfort and the predictable average can break down.
One house has a 0.5% fire chance — utterly unpredictable. Across 100,000 independent homes, an insurer expects about 500 fires a year, and the actual count lands close to 500 almost every year. The individual is a mystery; the pool is predictable.
The individual is unpredictable; the large pool is not — that is the whole trick of insurance.
The law promises the AVERAGE converges, not that outcomes 'balance out' — a run of heads is not 'due' to be corrected. And it relies on independence; correlated catastrophe risks do not obey it, which is why diversification has limits.