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  "body": "\n## Summary\n\nA binomial proportion confidence interval is a [[confidence interval|0S6T9KQNC0]] for the success probability $p$ of a [[binomial distribution|0WCWA1KT3M]].\nIt is computed from the number of successes in $n$ trials.\nBecause the data are discrete, no interval has coverage exactly $1 - \\alpha$ for every $p$.\nSo there are several standard intervals, such as the Wald, Wilson and Clopper–Pearson intervals, with different trade-offs between coverage and length.\n\n## Explanations\n\n### L1 human\n\nYou toss a coin $n$ times, see $x$ heads, and want a range of plausible values for the true probability of heads.\nThe simplest answer is the observed proportion plus or minus about two standard errors.\nFor $x = 40$ heads in $n = 100$ tosses, this gives roughly $0.40 \\pm 0.10$, so the interval $[0.30, 0.50]$ at level about $95\\%$.\nThis simple interval behaves badly when $p$ is near $0$ or $1$, which is why better intervals exist.\n",
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