2024/09/28 by Mathew Drexel, Drexel, Mathew, Xuanshan Peng +3
Social Sciences · #05A19 #60C05 #Combinatorics (math.CO) #FOS: Mathematics #Linguistic Variation and Morphology #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2409.19195
openalex publication_date 2024/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 This setup, commonly known as Penney's ante, was popularized by Conway, who found (in unpublished work) a simple formula for the probability that a given word occurs first.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 = (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 = (1)/(2) we characterize when a longer word can be statistically favorable, and for p ≠ (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 → 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.