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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A_1, A_2, A_3, \\dots$ be a finite or infinite sequence of [[events|00SQGJNRTB]].\nThen\n$$P\\left(\\bigcup_{i} A_i\\right) \\le \\sum_{i} P(A_i).$$\n\n## Remarks\n\nThe inequality is also called the union bound. It needs no hypothesis on the events: they need not be disjoint or independent.\nEquality holds when the events are pairwise disjoint.\nThe Bonferroni inequalities refine it with the terms of the [[inclusion–exclusion principle|0FF7726C7P]]. For example, $P\\left(\\bigcup_{i=1}^n A_i\\right) \\ge \\sum_i P(A_i) - \\sum_{i \u003c j} P(A_i \\cap A_j)$.\n\n## History\n\nThe inequality is named after George Boole.\n\n## Explanations\n\n### L1 human\n\nThe chance that at least one of several bad things happens is at most the sum of their chances.\nIf each of $10$ parts fails with probability $0.01$, then some part fails with probability at most $0.1$.\n",
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