1. Word length, bias and bijections in Penney's ante
- Author
-
Drexel, Mathew, Peng, Xuanshan, and Richey, Jacob
- Subjects
Mathematics - Combinatorics ,Mathematics - Probability ,60C05, 05A19 - 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 This setup, commonly known as Penney's ante, was popularized by Conway, who found (in unpublished work) a simple formula for the probability that a given word occurs first.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., Comment: 20 pages, 2 figures
- Published
- 2024