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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space and let $A, B, C \\in \\mathcal{F}$ with $P(C) \u003e 0$.\nThe events $A$ and $B$ are conditionally independent given $C$ if\n$$P(A \\cap B \\mid C) = P(A \\mid C) \\, P(B \\mid C),$$\nwhere $P(\\cdot \\mid C)$ is the [[conditional probability|0FJB06AD2P]] given $C$.\n\n## Remarks\n\nIf also $P(B \\cap C) \u003e 0$, the condition is equivalent to $P(A \\mid B \\cap C) = P(A \\mid C)$.\nConditional independence does not imply independence, and independence does not imply conditional independence.\nExample: choose one of two coins at random, a fair coin or a coin with two heads, and toss it two times. Given the coin, the two tosses are independent. Without this knowledge, they are dependent.\nFor random variables $X$ and $Y$ given $Z$, the definition uses the conditional distributions given $Z$.\n\n## Explanations\n\n### L1 human\n\nTwo results can be linked only through a hidden common cause.\nWhen you know the cause, the link disappears. This is conditional independence given the cause.\n",
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