First Passage Properties of the P\'olya Urn Process

Tibor Antal, E. Ben-Naim, and P.L. Krapivsky

We study first passage statistics of the P\'olya urn model. In this random process, the urn contains two types of balls. In each step, one ball is drawn randomly from the urn, and subsequently placed back into the urn together with an additional ball of the same type. We derive the probability $G_n$ that the two types of balls are equal in number, for the first time, when there is a total of $2n$ balls. This first passage probability decays algebraically, $G_n\sim n^{-2}$, when $n$ is large. We also derive the probability that a tie ever happens. This probability is between zero and one, so that a tie may occur in some realizations but not in others. The likelihood of a tie is appreciable only if the initial difference in the number balls is of the order of the square-root of the total number of balls.


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