How Many Rounds Should You Expect in Urn Solitaire?

TL;DR

This paper investigates the expected number of rounds in a specific urn sampling process, revealing its complexity and providing a sketch of the proof, contrasting it with simpler classical problems like gambler's ruin.

Contribution

It uncovers and sketches the proof of the complex expected duration of a sampling process with a constant probability, highlighting its difference from simpler classical problems.

Findings

Expected duration is highly complex and not easily expressed.

Probability remains constant at 1/2 despite complexity.

The process's expected duration exceeds simple analytical expressions.

Abstract

A certain sampling process, concerning an urn with balls of two colors, proposed in 1965 by B.E. Oakley and R.L. Perry, and discussed by Peter Winkler and Martin Gardner, that has an extremely simple answer for the probability, namely the constant function 1/2, has a far more complicated expected duration, that we discover and sketch the proof of. So unlike, for example, the classical gambler's ruin problem, for which both `probability of winning' and `expected duration' have very simple expressions, in this case the expected number of rounds is extremely complicated, and beyond the scope of humans.