Law of Large Numbers

The law of large numbers is a theorem in probability theory stating that as the number of independent trials of a random experiment increases, the average of the results obtained converges toward the experiment's true expected value. Swiss mathematician Jacob Bernoulli first proved a version of the theorem, which he called his Golden Theorem, in his 1713 posthumously published work Ars Conjectandi, establishing the mathematical justification for the long-observed intuition that the more times a random event is repeated, the more closely the proportion of outcomes will match the underlying probability. French mathematician Simeon Denis Poisson later gave the theorem its now-standard name in an 1835 publication, and it is now formally distinguished into a weak law, concerning convergence in probability, and a strong law, concerning almost-sure convergence. The theorem provides the theoretical basis for estimating a population's true proportions or averages from a sufficiently large random sample, and is distinct from the related central limit theorem, which describes the shape of the distribution of sample averages around the true value rather than merely their convergence to it.