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  "body": "\n## Statement\n\nLet $\\Gamma$ be a finite set, and let $R \\subseteq \\Gamma$ be a random subset that contains each $r \\in \\Gamma$ independently with probability $p_r$.\nLet $A_1, \\dots, A_n$ be subsets of $\\Gamma$, and let $B_i$ be the [[event|031TPJVS74]] that $A_i \\subseteq R$.\nWrite $i \\sim j$ if $i \\ne j$ and $A_i \\cap A_j \\ne \\emptyset$.\nLet\n$$\\mu = \\sum_{i=1}^n \\Pr(B_i), \\qquad \\Delta = \\sum_{(i, j) : i \\sim j} \\Pr(B_i \\cap B_j),$$\nwhere the second sum runs over ordered pairs.\nThen\n$$\\Pr\\Big(\\bigcap_{i=1}^n \\overline{B_i}\\Big) \\le e^{-\\mu + \\Delta/2}.$$\n\n## Remarks\n\nIf moreover $\\Delta \\ge \\mu$, a further argument gives the bound $e^{-\\mu^2/(2\\Delta)}$.\nIn the other direction, the [[Harris inequality|W1YH3YD3S4]] gives $\\Pr\\big(\\bigcap_i \\overline{B_i}\\big) \\ge \\prod_i (1 - \\Pr(B_i))$.\nSo when $\\Delta$ is small and each $\\Pr(B_i)$ is small, the probability that no $B_i$ occurs is close to $e^{-\\mu}$, as for independent events.\nFor example, in the [[random graph|TNT3DG0HRJ]] $G(n, p)$, take $\\Gamma$ to be the set of edges of $K_n$ and the $A_i$ to be the edge sets of triangles.\nThen $\\mu = \\binom{n}{3} p^3$.\nWhen $p = c/n$ for a constant $c \u003e 0$, one has $\\mu \\to c^3/6$ and $\\Delta \\to 0$ as $n \\to \\infty$.\nWith the lower bound, the probability that $G(n, p)$ has no triangle tends to $e^{-c^3/6}$.\nThe proof writes the probability as a product of the factors $\\Pr\\big(\\overline{B_i} \\mid \\bigcap_{j \u003c i} \\overline{B_j}\\big)$.\nIt bounds each conditional probability of $B_i$ from below, using the Harris inequality for increasing events.\n\n## Explanations\n\n### L1 human\n\nMany small \"bad\" patterns may appear in a random set.\nIf the patterns rarely overlap, the chance that none of them appears is close to what it would be if they were independent.\nThe Janson inequality makes this precise, with an error term that measures the overlaps.\n\n### L2 human\n\nThe Janson inequality is a standard tool in the theory of random graphs and the [[probabilistic method|3B55HNS754]].\nIt gives exponentially small upper bounds on the probability that a random graph contains no copy of a given subgraph.\nCompared with the second moment method, it gives a much sharper bound on the probability of the lower tail.\n",
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