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  "body": "\n## Definition\n\nLet $S$ be a nonempty [[finite set|0BBTH3YSZA]] with $n$ elements.\nThe discrete uniform distribution on $S$ is the distribution with the [[probability mass function|0WF4EJP91W]]\n$$p(s) = \\frac{1}{n} \\quad \\text{for } s \\in S.$$\nA random variable $X$ is uniform on $S$ if $P(X = s) = 1/n$ for each $s \\in S$.\n\n## Remarks\n\nThen $P(X \\in A) = |A| / n$ for each $A \\subseteq S$. This is the [[classical definition of probability|0AKNPYQ04R]].\nFor $S = \\{a, a + 1, \\dots, b\\}$ with integers $a \\le b$ and $n = b - a + 1$, the mean is $(a + b)/2$ and the variance is $(n^2 - 1)/12$.\nThe result of a fair die is uniform on $\\{1, \\dots, 6\\}$, with mean $7/2$.\n\n## Explanations\n\n### L1 human\n\nThe discrete uniform distribution is \"all outcomes equally likely\".\nA fair die, a fair coin and a well-shuffled deck all give uniform distributions.\n",
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