First Passage Properties of the Polya Urn Process

TL;DR

This paper analyzes the first passage properties of the Polya urn process, deriving probabilities related to when the two ball types first become equal and the likelihood of ties occurring.

Contribution

It provides analytical expressions for first passage probabilities and the conditions under which ties are likely in the Polya urn model.

Findings

First passage probability decays as n^{-2} for large n.

Probability of a tie is between zero and one, depending on initial conditions.

Ties are more likely when initial difference is of order sqrt of total balls.

Abstract

We study first passage statistics of the Polya 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 ~ 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.