the weak law of large numbers

Everyone has the intuition that 'averages settle down': flip a coin a thousand times and the fraction of heads is very likely near one half. The weak law of large numbers is the precise version of that gut feeling. It says the sample mean of many independent draws from the same distribution will, with high probability, sit close to the true mean — and the more draws you take, the higher that probability climbs.

Precisely: if X_1, X_2, ... are independent and identically distributed with finite mean mu, and you form the running average X-bar_n = (X_1 + ... + X_n)/n, then X-bar_n converges in probability to mu. Unpacked: for any tolerance epsilon, P(|X-bar_n - mu| > epsilon) approaches 0 as n grows. A clean proof when the variance sigma^2 is finite uses Chebyshev's inequality: Var(X-bar_n) = sigma^2/n, so the bound P(|X-bar_n - mu| > epsilon) <= sigma^2/(n epsilon^2) clearly goes to 0. (The full theorem needs only a finite mean, but this version makes the mechanism visible.)

Two honest clarifications. First, 'weak' refers to the MODE of convergence (in probability), not to the strength of the conclusion — the strong law upgrades the same statement to almost sure convergence. Second, the law is about the AVERAGE converging, not about sums 'evening out'. After a streak of heads the coin is not 'due' for tails; that is the gambler's fallacy. The average drifts toward one half not because past results are corrected, but because they are diluted by an ever-growing pile of fresh, indifferent trials.