We choose balls out of a box. In a box we have 2 white balls with numbers 1 and 2, three black balls with numbers 1,2,3. From a box without replacement we choose two balls. Let X denote biggest number chosen and Y denote number of white balls we have chosen. Find $\mathbb{E}(Y|X)$. I know i have to find $\mathbb{E}(Y|X=1)$,$\mathbb{E}(Y|X=2)$,$\mathbb{E}(Y|X=3)$. Then i need to set up a system of equations, but how to find those single conditional values?
EDIT: (still i have not received a satisfactory answer. Now i am looking for a solution for: $\mathbb{E}(X|Y)$. Thus:
$\mathbb{E}(X|Y=0)=0*1+\frac{1}{3}*2+\frac{2}{3}*3$
$\mathbb{E}(X|Y=1)=\frac{1}{6}*1+\frac{1}{2}*2+\frac{2}{6}*3$
$\mathbb{E}(X|Y=2)=2$
Is this attempt correct?