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  "body": "\n## Summary\n\nThe law of large numbers says that the average of many independent repetitions of a random experiment tends to the expected value.\nFor an [[i.i.d.|0AJN7WHBJZ]] sequence $X_1, X_2, \\dots$ with finite mean $\\mu$, the sample mean $\\bar{X}_n = (X_1 + \\dots + X_n)/n$ tends to $\\mu$.\nThe [[weak law|0BFDHF2H7N]] gives convergence in probability, and the [[strong law|0DFWHQ839T]] gives almost sure convergence.\n\n## Remarks\n\nThe law fails without a finite mean. For i.i.d. standard Cauchy variables, $\\bar{X}_n$ is standard Cauchy for each $n$ and does not converge.\nFor indicator variables of an event $A$, the law says that the relative frequency of $A$ tends to $P(A)$. This links probability with long-run frequency.\n\n## Explanations\n\n### L1 human\n\nIf you toss a fair coin many times, the fraction of heads comes close to $1/2$.\nThe law of large numbers is the precise form of this experience. It is the reason why averages of many measurements are reliable.\n",
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