·Mathematics & Probability
Section 1
The Core Idea
The law of large numbers says that as you repeat a random experiment more and more times, the average of the results converges to the expected value. Sample a few times and the average can be far from the true mean; sample thousands of times and the average gets close. The theorem was formalised by Jacob Bernoulli (1713) and refined by Kolmogorov; it's the reason casinos and insurers make money. They don't need to win every bet or every policy — they need enough volume so that the average outcome is predictable.
Variance gets averaged out. The house edge compounds; the law does the work.
The practical implication: small samples are noisy, large samples are stable. A startup with 10 customers can have a 90% NPS or a 20% NPS and neither is a reliable signal of the underlying satisfaction distribution. With 1,000 customers, the sample mean is a decent estimate. The same logic applies to A/B tests (run them long enough), to hiring (one great or bad hire doesn't define your process), and to investing (a few trades don't prove edge). When someone generalises from a handful of cases, the law of large numbers is being violated. When someone insists on more data before deciding, they're invoking it.
Two caveats. First, the law says the average converges; it doesn't say any single outcome is predictable. You can still have a terrible run in a fair game. Second, the law assumes independent (or weakly dependent) trials with a fixed distribution. If the process changes over time or trials are correlated, convergence can be slow or fail. Use the law to justify collecting more data when the sample is small and to be sceptical of conclusions drawn from few observations.
The law also explains why diversification works when bets are independent: the portfolio's average return converges to the weighted average of expected returns, and variance shrinks. When bets are correlated — same sector, same factor — the effective n is smaller and the law helps less. One bad event can hit the whole portfolio. So the law supports "many independent bets" but doesn't support "many correlated bets" as a way to reduce risk.
