I cannot seem to solve this conditional probability question.

Suppose $X$ and $Y$ are two events from a sample space with $\Pr(X) = 0.25$, $\Pr(Y) = 0.5$ and $\Pr(X|X \cup Y) = 0.5.$ Find $\Pr(X \cup Y)$.

I know that the union of $X$ and $Y$ will be:

$$ \Pr(X \cup Y) = \Pr(X) + \Pr(Y) - \Pr(X ∩ Y) $$

But I am having issues finding $(X ∩ Y)$. I think this is linked to Bayes Theorem in some way but I am going in circles.

  • 3
    $\begingroup$ You can deduce this from the pieces of information given. Try drawing a Venn diagram with some overlap of $X$ and $Y$; you should be able to figure out the formula for $P(X|X\cup Y)$ given $P(X)$, $P(Y)$, and one other term, then rearrange it to find the value of the other term, and go from there. $\endgroup$ – jbowman Sep 30 '18 at 21:55

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