Many rolls settle the average—streaks do not “owe” a correction
The law of large numbers says sample averages of independent, identically distributed draws converge to the true mean when that mean exists. Casinos rely on it; gamblers’ fallacies deny it. Heavy-tailed laws without a finite mean can refuse to settle at all.
Formally, for i.i.d. samples the sample mean converges to the population mean. That guarantee of stable long-run averages underpins statistics, economics, and insurance. A casino may lose on one roulette spin yet tend toward a predictable edge over many spins. The law speaks only about large samples: a short run need not sit near the expectation, and a streak is not owed an immediate balancing outcome—that is the gambler’s fallacy. Monte Carlo methods lean on the same idea: more random draws usually improve the approximation when the expectation exists.
A fair die’s faces average to 3.5; many rolls push the sample mean toward that value. Relative frequency in Bernoulli trials likewise approaches the success probability—so a fair coin’s heads proportion almost surely approaches one half. Even then, the absolute gap between heads and tails counts typically grows, while that gap divided by the number of flips shrinks toward zero. Failures matter too: averages from a Cauchy law, or Pareto with shape below one, need not converge because the mean is missing or infinite. Selection bias also survives more trials; the law does not wash it away.
History runs from Gerolamo Cardano’s unproved intuition through Jacob Bernoulli’s binary “golden theorem,” published in Ars Conjectandi in 1713 after more than twenty years’ work. S. D. Poisson named “la loi des grands nombres” in 1837. Later refinements by Chebyshev, Markov, Borel, Cantelli, Kolmogorov, and Khinchin split the story into weak and strong forms. The weak law (Khinchin’s) is convergence in probability; the strong law (Kolmogorov’s) is almost-sure convergence and implies the weak. Khinchin showed in 1929 that for i.i.d. variables a finite expectation suffices for the weak law; Kolmogorov proved related strong-law results in 1930 and 1933. Finite variance speeds proofs but is not required for the law itself.
Source: Law of large numbers
