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  "body": "\n## Statement\n\nLet $X_1, \\dots, X_n$ be [[independent|0XCE070V8B]] real random variables with $a_i \\le X_i \\le b_i$ almost surely, where $a_i \u003c b_i$ are reals. Let $S_n = X_1 + \\dots + X_n$.\nThen for each $t \u003e 0$,\n$$P\\left(S_n - E[S_n] \\ge t\\right) \\le \\exp\\left(-\\frac{2t^2}{\\sum_{i=1}^n (b_i - a_i)^2}\\right).$$\n\n## Remarks\n\nThe same bound holds for $P(S_n - E[S_n] \\le -t)$. So $P(|S_n - E[S_n]| \\ge t) \\le 2 \\exp\\left(-2t^2 / \\sum_i (b_i - a_i)^2\\right)$.\nThe proof uses the [[Chernoff bound|014PEBGDYF]] together with Hoeffding's lemma: $E[e^{s(X - E[X])}] \\le e^{s^2 (b - a)^2 / 8}$ if $a \\le X \\le b$.\nFor $n$ independent tosses of a coin with $P(\\text{heads}) = p$, the fraction $H$ of heads satisfies $P(|H - p| \\ge \\varepsilon) \\le 2 e^{-2n\\varepsilon^2}$.\n\n## History\n\nThe inequality is named after Wassily Hoeffding.\n\n## Explanations\n\n### L1 human\n\nA sum of many independent bounded terms is very unlikely to be far from its mean.\nThe probability of a large deviation decreases like $e^{-c t^2}$. This is why polls with a few thousand people give reliable results.\n",
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