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We study first passage statistics of the Polya urn model. In this randomprocess, the urn contains two types of balls. In each step, one ball is drawnrandomly from the urn, and subsequently placed back into the urn together withan additional ball of the same type. We derive the probability G_n that the twotypes of balls are equal in number, for the first time, when there is a totalof 2n balls. This first passage probability decays algebraically, G_n ~ n^{-2},when n is large. We also derive the probability that a tie ever happens. Thisprobability is between zero and one, so that a tie may occur in somerealizations but not in others. The likelihood of a tie is appreciable only ifthe initial difference in the number balls is of the order of the square-rootof the total number of balls.

First Passage Properties of the Polya Urn Process