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The probability that an event A will occur, when another event B is known to occur or to have occurred. It is commonly denoted by P(A|B).

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Does independence of conditional variable and random variable imply independence of reverse ...

Assuming homework: First, you can even try make this work without the hard part, the conditionals: what if $X$ and $Y$ were independent? Then $X \mid Y = X$, and $Y \mid X = Y$. Now a counterexample …
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