If I am given two events $A$ and $B$ and the probabilities $P(A), P(A|B), P(B), P(B|A)$, and am told that the random variables $X$ and $Y$ are defined as
$X(a) = 1\ \text{if}\ a \in A\text{, else}\ 0$ $Y(b) = 1\ \text{if}\ b \in B\text{, else}\ 0$
how can I then go about to calculate the expected values of these variables? What probability density functions should I use for $p(a \in A)$ and $p(b \in B)$, given that none are specified explicitly?