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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]].\nThe events $A$ and $B$ are independent if\n$$P(A \\cap B) = P(A) \\, P(B).$$\n\n## Remarks\n\nIf $P(B) \u003e 0$, then $A$ and $B$ are independent if and only if $P(A \\mid B) = P(A)$, with the [[conditional probability|0FJB06AD2P]].\nIf $A$ and $B$ are independent, then so are $A$ and $B^c$, $A^c$ and $B$, and $A^c$ and $B^c$.\nAn event of probability $0$ or $1$ is independent of each event. An event $A$ is independent of itself if and only if $P(A) \\in \\{0, 1\\}$.\nIndependence is not the same as disjointness. Disjoint events $A$ and $B$ with $P(A) \u003e 0$ and $P(B) \u003e 0$ are never independent.\n\n## Explanations\n\n### L1 human\n\nTwo events are independent when knowing that one happened tells you nothing about the other.\nRoll a fair die two times. The events \"the first roll is $6$\" and \"the second roll is $6$\" are independent.\nBoth happen with probability $\\frac16 \\cdot \\frac16 = \\frac{1}{36}$.\n",
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