A theorem stating that as the number of independent trials increases, the average of the results gets closer to the expected value.
Card 1 of 259. Answer: A theorem stating that as the number of independent trials increases, the average of the results gets closer to the expected value.
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The law of large numbers says that the average of many independent draws of the same random variable gets closer to its expected value as the number of draws grows. It needs only a finite expected value: the weak law gives convergence in probability, the strong law convergence almost surely.
For a fair coin, 7 heads in 10 flips is unremarkable. After 10,000 flips the standard deviation of the proportion of heads is 0.5/√10,000 = 0.005, so about 95% of the time it lands between 0.49 and 0.51. Jacob Bernoulli proved this case in Ars Conjectandi, published in 1713 after his death; Siméon Denis Poisson named the law in 1837.
The classic follow-up is the gambler's fallacy. The law does not say a run of tails will be balanced by extra heads: the gap between the count of heads and n/2 typically grows, like √n, and only the proportion settles. The central limit theorem then describes the shape of the fluctuations around the expected value.
