Extremal poker hand rankings: why the standard 52 card deck and a 3044 card deck are special
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914143901057024 |
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| author | Williamson, Christopher |
| author_facet | Williamson, Christopher |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06145 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal poker hand rankings: why the standard 52 card deck and a 3044 card deck are special Williamson, Christopher History and Overview Combinatorics 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. |
| title | Extremal poker hand rankings: why the standard 52 card deck and a 3044 card deck are special |
| topic | History and Overview Combinatorics |
| url | https://arxiv.org/abs/2511.06145 |