Calculate $E[XY]$ of dependent variables

Question

I'm having a little trouble whit a probabilistic exercise.

The problem says this: There's a vase whit 10 marbles, 4 black and 6 white. Now I extract 2 of them without reposition. Being $X,Y$ random variables, such: $X=\begin{Bmatrix} 1 \text{ if the number of white marbles is even} \\ 0 \text{ if the number of white marbles is odd} \end{Bmatrix}$

$Y= \text{ number of black marbles extracted}$

Then $Y \sim \mathcal{Hypergeometric} (3,10,6)$

And

$P(X=1)=P(Y=1)+P(Y=3)= 7/15$

$P(X=0) = 1- P(X=1) = 8/15$

Now I'm supposed to calculate $E[XY]$ but I'm not sure how.

As a guess, I think calculating $P(X=x,Y=y)$ for all $x,y \in \{0,1,2\}$

And then do somthing like $E[XY]=\suṃ_{x=0}^3\suṃ_{y=0}^3X.Y.P(X=x,Y=y)$ would give me the answer.

Answer 1

Hint: Express $X$ as a function of $Y$: $$ X = 1_{Y\text{ is even}} $$