{
  "aliases": [
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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A \\in \\mathcal{F}$ be an [[event|00SQGJNRTB]].\nLet $A^c = \\Omega \\setminus A$ be the [[complement|0B1FSS1VZW]] of $A$.\nThen\n$$P(A^c) = 1 - P(A).$$\n\n## Remarks\n\nThe proof: $A$ and $A^c$ are disjoint and $A \\cup A^c = \\Omega$, so $P(A) + P(A^c) = P(\\Omega) = 1$ by additivity.\nWith $A = \\Omega$, the rule gives $P(\\varnothing) = 0$.\nThe rule is useful when $A^c$ is simpler than $A$. For example, the probability of at least one six in four rolls of a fair die is $1 - (5/6)^4 \\approx 0.518$.\n\n## Explanations\n\n### L1 human\n\nEither $A$ happens or it does not, and the two chances add up to $1$.\nIf the chance of rain is $0.3$, then the chance of no rain is $0.7$.\n",
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  "name": "Probability of the complement",
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