X,Y events with P(X Y)=P(X Y).

Question

$X,Y$ events with $P(X\cup Y)=P(X\cap Y)$. Statement 1: $P(X\cap Y')=P(X'\cap Y)=0$. Statement 2: $P(X)+P(Y)=2P(X\cap Y)$.

Accepted Answer

From $P(X\cup Y)=P(X)+P(Y)-P(X\cap Y)$ combined with the given equality, $P(X)+P(Y)=2P(X\cap Y)$ (Statement 2 true). Then $P(X\cap Y')=P(X)-P(X\cap Y)$ and $P(X'\cap Y)=P(Y)-P(X\cap Y)$; using Statement 2 these both equal 0, so Statement 1 is also true (but as a separate derivation, not a "correct explanation" relationship).

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