How to Answer Questions of the Type: If you toss a coin n times, how likely is HH to show up more than HT?

TL;DR

This paper demonstrates how symbolic computation, especially the Almkvist-Zeilberger algorithm, can effectively analyze probabilities in coin-tossing questions involving complex pattern counts.

Contribution

It introduces a novel application of the Almkvist-Zeilberger algorithm to solve probability questions related to pattern occurrences in coin toss sequences.

Findings

Successfully applied symbolic computation to coin-toss probability problems

Provided a method to determine likelihoods of specific pattern counts in sequences

Extended analysis to more general pattern-based questions

Abstract

On March 16, 2024, Daniel Litt, in an X-post, proposed the following brainteaser: "Flip a fair coin 100 times. It gives a sequence of heads (H) and tails (T). For each HH in the sequence of flips, Alice gets a point; for each HT, Bob does, so e.g. for the sequence THHHT Alice gets 2 points and Bob gets 1 point. Who is most likely to win?" We show the power of symbolic computation, in particular the (continuous) Almkvist-Zeilberger algorithm, to answer this, and far more general, questions of this kind.