First-passage properties of the Pólya urn process

Abstract

We study first-passage statistics of the Pólya urn model. In this random process, the urn containsballs of two types. In each step, one ball is drawn randomly from the urn, and subsequently placedback into the urn together with an additional ball of the same type. We derive the probabilityGn that the balls of the two types become equal in number, for the first time, when there are a total of2n balls. This first-passage probability decays algebraically,Gn ∼ n − 2, whenn is large. We also derive the probability that a tie ever happens. This probability is betweenzero and one, so a tie may occur in some realizations but not in others. The likelihood of atie is appreciable only if the initial difference in the number of balls is of the order of thesquare root of the total number of balls.