law of large numbers: a standalone construct

Jacob Bernoulli first proved the law of large numbers in his posthumous Ars Conjectandi (1713), showing that as the number of independent trials of a chance experiment grows, the observed proportion of a given outcome converges toward its true underlying probability; Siméon Denis Poisson gave the result its lasting name, la loi des grands nombres, in 1837. The modern form, due chiefly to Andrey Kolmogorov in the 1930s within his measure-theoretic foundation of probability, distinguishes the weak law, in which the sample average converges in probability to the expected value, from the strong law, in which the average converges to the expected value with probability one. In plain terms the law says that the arithmetic mean of many independent measurements of the same quantity settles near that quantity's expected value, and that averaging over more observations makes large departures from that value increasingly unlikely. What makes the law a landmark of probability theory is its generality: it holds for essentially any quantity with a finite expected value regardless of the shape of its distribution, and it supplies the formal justification for estimating an unknown mean, probability, or rate by averaging repeated observations. As a separate cross-reference, the phrase is sometimes loosely invoked in everyday business talk to describe how a very large firm finds it harder to sustain a high growth rate, a usage distinct from the probability theorem and not treated as a construct here.[1][2][3][4]

Unit

A limit theorem stating that the sample mean converges to the expected value as the number of observations grows

Recovery means

Not applicable

Measured by

Stated and proved as a limit theorem; illustrated empirically by simulation, where sample means are seen to stabilise near the expected value as the number of trials increases