{"id":"law-of-large-numbers","domain":"probability-statistics","type":"Theorem","title":"Law of Large Numbers","slug":"law-of-large-numbers","url":"/mathematics/probability-statistics/law-of-large-numbers/","proved":"1713 (weak version), Jacob Bernoulli; 1930s (strong version), Kolmogorov","summary":"Sample average converges to true expected value as sample size grows. Foundation of insurance, casinos, polling. Not to be confused with gambler's fallacy.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Feller, W. (1968). An Introduction to Probability Theory. Vol. 1, 3rd ed."},{"tier":1,"citation":"Billingsley, P. (1995). Probability and Measure. 3rd ed."},{"tier":2,"citation":"Stigler, S.M. (1986). The History of Statistics. Harvard University Press."},{"tier":3,"citation":"Tijms, H. (2007). Understanding Probability. 2nd ed."}],"relationships":[{"type":"built_from","target":"random-variable","label":"Random Variable"},{"type":"complements","target":"central-limit-theorem","label":"Central Limit Theorem"},{"type":"foundational_to","target":"normal-distribution","label":"Normal Distribution"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.9"}