Word length, bias and bijections in Penney's ante

TL;DR

This paper analyzes win probabilities in Penney's ante, exploring biases, symmetries, and optimal word pairs for sequences over a binary alphabet, extending previous results and providing new combinatorial bijections.

Contribution

It extends analysis of Penney's ante to arbitrary word lengths and biases, characterizes when longer words are favorable, and introduces explicit bijections explaining symmetries.

Findings

Bounds on win probabilities for longer words.

Characterization of favorable longer words at p=1/2.

Conjecture on optimal pairs for p ≠ 1/2.

Abstract

Fix two words over the binary alphabet $\{0,1\}$, and generate iid Bernoulli$(p)$ bits until one of the words occurs in sequence. This setup, commonly known as Penney's ante, was popularized by Conway, who found (in unpublished work) a simple formula for the probability that a given word occurs first. We study win probabilities in Penney's ante from an analytic and combinatorial perspective, building on previous results for the case $p = \frac{1}{2}$ and words of the same length. For words of arbitrary lengths, our results bound how large the win probability can be for the longer word. When $p = \frac{1}{2}$ we characterize when a longer word can be statistically favorable, and for $p \neq \frac{1}{2}$ we present a conjecture describing the optimal pairs, which is supported by computer computations. Additionally, we find that Penney's ante often exhibits symmetry under the transformation $p \to 1-p$. We construct new explicit bijections that account for these symmetries, under conditions that can be easily verified by examining auto- and cross-correlations of the words.