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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0ZYBZ34RT2]] and let $A, B \\in \\mathcal{F}$ be [[events|00SQGJNRTB]] with $P(B) \u003e 0$.\nThe conditional probability of $A$ given $B$ is\n$$P(A \\mid B) = \\frac{P(A \\cap B)}{P(B)}.$$\n\n## Remarks\n\nFor fixed $B$ with $P(B) \u003e 0$, the map $A \\mapsto P(A \\mid B)$ is a [[probability measure|0VGMEBVW1M]] on $(\\Omega, \\mathcal{F})$. It gives all its mass to $B$.\nFor one roll of a fair die, $P(\\{6\\} \\mid \\text{even}) = (1/6) / (1/2) = 1/3$.\nIn general $P(A \\mid B) \\ne P(B \\mid A)$. [[Bayes' theorem|0NM6E9VW04]] relates the two.\nThe definition does not apply when $P(B) = 0$. Conditioning on such events needs the [[conditional expectation|0YAMJRK04E]].\n\n## Explanations\n\n### L1 human\n\nOnce you know that $B$ happened, the outcomes outside $B$ are no longer possible.\nSo you keep only the part of $A$ inside $B$, and you rescale by $P(B)$ so that the total is $1$ again.\n",
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