[Submitted on 28 Mar 2019 (v1), last revised 31 May 2026 (this version, v3)]

View PDF

Abstract:Penney's game is a two-player zero-sum game in which each player chooses a three-flip pattern of heads and tails, and the winner is the player whose pattern occurs first in repeated tosses of a fair coin. Because the players choose sequentially, the second mover has the advantage. In fact, for any three-flip pattern, there is another three-flip pattern that is strictly more likely to occur first. This paper provides a novel no-arbitrage argument that generates the winning odds corresponding to any pair of distinct patterns. The resulting formula is equivalent to that generated by Conway's ``leading number'' algorithm. The accompanying betting-odds intuition adds insight into why Conway's algorithm works. The proof is simple and easy to generalize to games involving more than two outcomes, unequal probabilities, and competing patterns of various lengths. Additional results on the expected duration of Penney's game are presented.

Submission history

From: Joshua Benjamin Miller [view email]
[v1] Thu, 28 Mar 2019 18:01:15 UTC (779 KB)
[v2] Tue, 23 Apr 2019 09:56:55 UTC (384 KB)
[v3] Sun, 31 May 2026 15:00:54 UTC (613 KB)

Current browse context:

math.OC

Bookmark

Bibliographic and Citation Tools

Code, Data and Media Associated with this Article

Demos

Recommenders and Search Tools

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.