{
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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \\in \\mathcal{F}$ be [[events|00SQGJNRTB]].\nThen\n$$P(A \\cup B) + P(A \\cap B) = P(A) + P(B).$$\nEquivalently, $P(A \\cup B) = P(A) + P(B) - P(A \\cap B)$.\n\n## Remarks\n\nIf $A$ and $B$ are mutually exclusive, then $P(A \\cap B) = 0$ and $P(A \\cup B) = P(A) + P(B)$.\nThe rule is the probabilistic form of [[inclusion–exclusion for two sets|0C4YEQ9H5E]]. The [[inclusion–exclusion principle|0FF7726C7P]] gives the version for $n$ events.\nA consequence is $P(A \\cup B) \\le P(A) + P(B)$, the case $n = 2$ of [[Boole's inequality|0EGSEE9J9R]].\n\n## Explanations\n\n### L1 human\n\nWhen you add $P(A)$ and $P(B)$, you count the outcomes in both events twice, so you subtract $P(A \\cap B)$ once.\nFor one card from a deck, $P(\\text{heart or king}) = 13/52 + 4/52 - 1/52 = 16/52$.\n",
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