Word length, bias and bijections in Penney's ante

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 This setup, commonly known as Penney's ante, was popularized by Conway, who found (in unpublished work) a simple formula for the probability that a given word occurs first.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.