Assume there are $5$ balls in a box, $X$ black balls and $5-X$ white balls. $X$ is distributed uniform in {0,1,2,3,4,5}. We take a ball out of it, record the color, and put it back in the box, repeat 10 times, and record the number of black balls as random variable $Y$. Now what is $E(X|Y)$?
I know we can use the definition of conditional expect value and calculat it step by step. But it's a little complicated. Is there a more subtle way to solve this problem?