Paper

Extremal poker hand rankings: why the standard 52 card deck and a 3044 card deck are special

Williamson, Christopher

2025

Overview

We study poker hand rankings in the partially generalised setting of a deck with \\(r\\) ranks, rather than the typical 13 ranks. We provide the hand rankings for all \\(r\\) and observe some interesting phenomena such as the smallest \\(r\\) such that flushes rank below one-pair hands. Perhaps surprisingly, as \\(r\\) grows without bound, the hand ranking is not stable until \\(r=761\\) (a 3044 card deck). We consider showdown frequency, which is the frequency that a given type of hand is declared by a player at showdown, and make note of counterintuitive instances in which a hand with lower absolute frequency than some other hand nonetheless has a higher showdown frequency. This can be interpreted as a form of Gadbois paradox but in the typical setting of poker without wild cards. Conveniently, the standard deck with 13 ranks turns out to be the smallest deck that avoids a discrepancy between absolute frequency and showdown frequency for all hand types other than having a high card.

• Publisher: Cornell University Library, arXiv.org
• Subject: Decks