A Stochastic Urn Model for the Generalized Negative Binomial Distribution

We discuss a stochastic urn model in which there are two urns A and B. B is originally empty and A contains some fixed number of white and black balls. A player selects integers n > 0 and β ⪴0. Balls are drawn with replacement in A and balls of the same color are put in B as long as the number of white balls in B exceeds (β−1) times the number of black balls in B. Under this condition, the player stops after drawing n +βx balls and is declared to be a winner if urn B has x black balls. This number of black balls, x, is shown to have the generalized negative binomial distribution.