A proof that HT is more likely to outnumber HH than vice versa in a sequence of n coin flips

TL;DR

This paper proves that in a sequence of fair coin flips, Bob is more likely to win than Alice for all sequences of length n>=3, supported by an asymptotic analysis and efficient algorithms.

Contribution

It provides a formal proof that Bob's probability of winning exceeds Alice's for all n>=3, along with asymptotic analysis and computational methods.

Findings

Bob wins more often than Alice for all n>=3

Derived the asymptotic difference in win probabilities

Developed efficient algorithms for calculating these probabilities

Abstract

Consider the following probability puzzle: A fair coin is flipped n times. For each HT in the resulting sequence, Bob gets a point, and for each HH Alice gets a point. Who is more likely to win? We provide a proof that Bob wins more often for every n>=3. As a byproduct, we derive the asymptotic form of the difference in win probabilities, and obtain an efficient algorithms for their calculation.