Possible generalization of multivariate hypergeometric distribution

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?

• Did you at least take a look at the wiki section? Not to mention all the other Google hits. – Lee David Chung Lin Feb 25 at 8:38
• @LeeDavidChungLin Thanks for the suggestion. Yes, I did. I can only find examples when each ball belongs to a single colour category. I am trying to find related results when balls can belong to multiple categories. – Ben Feb 25 at 10:07
• The multivariate hypergeometric distribution allows us to compute the joint probability mass function for the $2^p$ category combinations but not, without some brute-force summation, the marginal probability mass function for the $p$ colour counts. – Ben Feb 25 at 10:18
• I see. Sorry I misunderstood your question. This is not an easy problem and at this point I don't have much to contribute. – Lee David Chung Lin Feb 25 at 11:57