I have got stuck with a problem related to the multivariate hypergeometric distribution. I pick $n$ balls from an urn containing a total of $N$ balls. Each ball is painted with (possibly multiple) coloured dots. Eventually I'd like to be able to deal with $p$ possible colours, but for the time being say $p=3$ and that the colours are red, green and blue. I have a big table with the total number of balls with each of the $2^p$ dot combinations.
I want to compute the joint probability for ending up with $k_r$ balls with a red dot, $k_g$ balls with a green dot and $k_b$ balls with a blue dot. To me it looks like the multivariate hypergeometric distribution but with overlapping ball categories. Has anyone seen this problem before? Could you please point me in a promising direction?