I am wondering if there exists a model for the probability of choosing at least k successful marbles from an urn of a (successful) and b marbles with replacement given n draws, where if an a marble is selected, it is replaced as a b marble but not vice versa (i.e., a marbles become b marbles if selected, but b marbles do not become a marbles).
I have come across beta distributions and beta-binomial distributions, but these don't seem to be exactly what I need.
Any help is much appreciated.