I've come across a process which is pretty simple to work with on a simulation basis, but I don't know any analytic solutions for it, and wondered if anyone knows what this distribution is and if they could direct me towards any papers/research or just a name/phrase to googleImagine this process if we have k holes each that can only accept 1 ball, and initially all holes are empty. We now randomly place a single ball in one of the holes with probability p_i (each hole having a different probability). We repeat the process for a second ball, however if we were going to put it a hole that was already filled we reject and try again, until it goes in an empty hole (or in other words, we renormalise the probabilities p_i removing the holes that are filled). We do this n < k times.I'm interested in the probability that after n trials a particular hole i will have a ball in it. If all the p_i are equal then this is clearly trivial, but with different p_i I'm getting no-where...

Ironically after a couple of days of looking for this and then deciding to post, I've just read about the Multivariate Wallenius' Noncentral Hypergeometric Distribution, which looks like this might be a special case of.Any one know anything about this?