There are $2N$ balls in an urn, 2 from each of N colors. We draw $2M$ (M < N) balls from the urn (independently and without replacement) and randomly place $M$ balls in each of two smaller urns. Let $X$ be the number of colors that are represented in both of the smaller urns.
What is the pmf of $X$?
Ideally, I'd like to know the pmf of X. If this is not tractable, then some way to estimate the tail of the distribution would be okay. In particular I am interested in this problem with values of N $\approx$ 1000 and $M \approx$ 50.
Example of the sampling procedure
Suppose $N = 5$ and $M = 3$. The big urn contains: RED, RED, BLUE, BLUE, GREEN, GREEN, PINK, PINK, YELLOW, YELLOW.
We draw RED, GREEN, BLUE, GREEN, PINK, YELLOW
and place them into:
small urn 1: RED, BLUE, GREEN
small urn 2: GREEN, PINK, YELLOW
Then $X$ = 1 since GREEN is represented in both of the smaller urns.
However, given the same original draw, if the balls had been placed as
small urn 1: RED, GREEN, GREEN
small urn 2: BLUE, PINK, YELLOW
Then $X$ = 0 because there is no color represented in both of the smaller urns.