Conditional probability of two intersected events If X and Y two independent events, and Z is a condition that determines the probabilities of X and Y, does $P(X\cap Y\mid Z) = P(X\mid Z) \cdot P(Y\mid Z)$?
For example, X could be the event of flipping a coin twice and the first time landing on heads, while Y could be the event flipping a coin twice and the second time landing on tails. Then Z could be a condition that states the fairness of the given coin. So $P(X\mid Z) = .5$ and $P(Y\mid Z) = .5$, while $P(X\mid Z^{c})=.8$ and $P(Y\mid Z) = .2$

Independence does not imply conditional independence. Let $X$ and $Y$ be the event that independent coin flips are heads and $Z$ be the event that either of them is heads (i.e. $Z = X\cup Y$.) Then $P(X|Z) = P(Y|Z) = \frac{2}{3}$ and $P(X\cap Y|Z) = 1/3.$
(Nor does conditional independence imply independence.)
