# Law of Large Numbers

**Type:** Theorem  
**Domain:** Probability & Statistics  
**Proved:** 1713 (weak), Jacob Bernoulli; 1930s (strong), Kolmogorov  
**Codex URL:** /mathematics/probability-statistics/law-of-large-numbers/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Sample average converges to true expected value as sample size grows. Foundation of insurance, casinos, polling.

## Weak Law
X̄ₙ converges to μ in probability. Provable via Chebyshev's inequality: P(|X̄ₙ−μ|≥ε) ≤ σ²/(nε²) → 0.

## Strong Law
X̄ₙ converges to μ almost surely (probability 1) — Kolmogorov, 1930s, requires measure theory.

## Gambler's fallacy
Common misconception: after 5 heads in a row, tails is NOT "due." Each flip remains independent; the law describes long-run behavior, not short-term correction.

## When it fails
Requires finite mean (and often finite variance). Cauchy distribution: sample average never converges, regardless of sample size.

## Applications
Insurance, opinion polling, quality control, Monte Carlo simulation methods.

## Sources
### Tier 1
- Feller, W. (1968). *An Introduction to Probability Theory*. Vol. 1.
- Billingsley, P. (1995). *Probability and Measure*. 3rd ed.
### Tier 2
- Stigler, S.M. (1986). *The History of Statistics*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*