Alice and Bob on $\Bbb X$: reversal, coupling, renewal

TL;DR

This paper investigates a coin-flip game involving Alice and Bob, using advanced probabilistic techniques like reversal, coupling, and renewal to determine the likely winner in a scenario that sparked social mathematical debate.

Contribution

It introduces a novel application of reversal, coupling, and renewal methods to analyze a probabilistic game involving sequences of coin flips.

Findings

Alice and Bob have nearly equal chances of winning.

The game outcome depends on the intricate structure of consecutive flips.

Probabilistic techniques effectively resolve intuitive conflicts in the game analysis.

Abstract

A neat question involving coin flips surfaced on $\Bbb X$, and generated an intensive `storm' of `social mathematics'. In a sequence of flips of a fair coin, Alice wins a point at each appearance of two consecutive heads, and Bob wins a point whenever a head is followed immediately by a tail. Who is more likely to win the game? The subsequent discussion illustrated conflicting intuitions, and concluded with the correct answer (it is a close thing). It is explained here why the context of the question is interesting and how it may be answered in a quantitative manner using the probabilistic techniques of reversal, coupling, and renewal.