The equalization probability of the Polya urn

TL;DR

This paper analyzes the probability that a Polya urn starting with more black than white balls will ever have equal numbers, linking it to a coin toss probability, providing a precise calculation.

Contribution

It derives an exact formula for the equalization probability of a Polya urn, connecting it to classical coin toss probabilities.

Findings

Equalization probability is twice the probability of limited heads in coin tosses.

Provides a closed-form expression for the probability of the urn ever reaching equal counts.

Establishes a novel connection between urn processes and coin toss probabilities.

Abstract

We consider a Polya urn, started with b black and w white balls, where b>w. We compute the probability that there are ever the same number of black and white balls in the urn, and show that it is twice the probability of getting no more than w-1 heads in b+w-1 tosses of a fair coin.