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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer and let $p \\in [0, 1]$.\nThe binomial distribution with parameters $n$ and $p$ is the distribution on $\\{0, 1, \\dots, n\\}$ with the [[probability mass function|0WF4EJP91W]]\n$$p(k) = \\binom{n}{k} p^k (1 - p)^{n-k} \\quad \\text{for } k = 0, 1, \\dots, n.$$\nHere $\\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]]. One writes $X \\sim \\operatorname{B}(n, p)$ or $X \\sim \\operatorname{Bin}(n, p)$.\n\n## Remarks\n\nThe values add up to $1$ by the [[binomial theorem|0EHWPWGHB2]]: $\\sum_{k} \\binom{n}{k} p^k (1 - p)^{n-k} = (p + 1 - p)^n = 1$.\nThe number of successes in $n$ independent [[Bernoulli trials|0PHC9FE2V5]] with success probability $p$ has this distribution. The factor $\\binom{n}{k}$ counts the positions of the $k$ successes.\nThe mean is $np$ and the variance is $np(1 - p)$.\nThe probability of exactly $2$ heads in $4$ tosses of a fair coin is $\\binom{4}{2} / 2^4 = 6/16$.\n\n## Explanations\n\n### L1 human\n\nRepeat a yes-or-no experiment $n$ times, independently, and count the yes answers.\nThe count has the binomial distribution. Each sequence with $k$ yes answers has probability $p^k (1 - p)^{n-k}$, and there are $\\binom{n}{k}$ such sequences.\n",
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