If X~HGeom(w,b,n), and Y = $X\choose2$, then:
$$E(Y) = \frac{{w \choose 2}{n \choose 2}}{{w+b} \choose 2}$$
This post: https://math.stackexchange.com/a/1669384/152816,
has a very detailed mathematical proof.
But since the result is very symmetric, it seems there could be an intuitive explanation of E(Y), i.e. we could story proof this E(Y) formula without complicated mathematical proof, could anyone think of an intuitive explanation of E(Y)?
Thanks.
$\binom{X}2$ is equal to the number of pairs of white balls which are chosen. For each particular pair of white balls, the probability they are chosen is equal to $$ \frac{\binom{b+w-2}{n-2}}{\binom{b+w}{n}}=\frac{n}{b+w}\cdot\frac{n-1}{b+w-1}:=p $$ Since there are $\binom{w}2$ pairs of white balls, and each is chosen about $p$ of the time, you would expect there to be $p\cdot \binom{w}2 $ pairs of white balls chosen. This implies the expected value of $\binom{X}2$ is $$ \frac{n}{b+w}\cdot\frac{n-1}{b+w-1}\cdot\binom{w}2=\frac{\binom{n}2\binom{w}2}{\binom{b+w}2}=E\Big[\binom{X}2\Big]. $$