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  "body": "\n## Definition\n\nLet $N$, $K$ and $n$ be integers with $N \\ge 1$, $0 \\le K \\le N$ and $0 \\le n \\le N$.\nThe hypergeometric distribution with parameters $N$, $K$ and $n$ has the [[probability mass function|0WF4EJP91W]]\n$$p(k) = \\frac{\\binom{K}{k} \\binom{N - K}{n - k}}{\\binom{N}{n}} \\quad \\text{for } \\max(0, n + K - N) \\le k \\le \\min(n, K).$$\n\n## Remarks\n\nA population has $N$ objects, and $K$ of them are successes. Draw $n$ objects without replacement, each subset of size $n$ equally likely. Then the number of successes in the sample has this distribution.\nThe mean is $nK/N$. The variance is $n \\frac{K}{N} \\frac{N - K}{N} \\frac{N - n}{N - 1}$ for $N \\ge 2$.\nWhen $N$ is large compared with $n$, the distribution is close to the [[binomial distribution|0VRWZDWW6B]] with parameters $n$ and $K/N$.\nThe probability of exactly $2$ aces in a hand of $5$ cards is $\\binom{4}{2} \\binom{48}{3} / \\binom{52}{5}$.\n\n## Explanations\n\n### L1 human\n\nAn urn has red and blue balls. Draw a few balls without putting them back, and count the red ones.\nThe count has the hypergeometric distribution. Each draw changes the urn, so the draws are not independent.\n",
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