# When Does A Straight Beat A Full House?

## Session Title

Session 3-1-D: Mathematics and Statistics II

## Presentation Type

Paper Presentation

## Location

Park MGM, Las Vegas, NV

## Start Date

25-5-2023 9:00 AM

## End Date

25-5-2023 10:30 AM

## Disciplines

Discrete Mathematics and Combinatorics | Probability

## Abstract

In 5-card poker played with the standard 52-card deck, full houses beat flushes, which beat straights. In this paper, we look at how alternate decks with different numbers of suits or ranks lead to different orderings of these 3 hands. We find that all 6 orderings of the 3 hands are possible with decks of fewer than 200 cards and that the standard deck sits at a precarious point: varying either the 4 suits or 13 ranks often leads to a deck where the usual order is no longer correct.

Implications: For possible poker games played with nonstandard decks, changing the deck can change the game; we outline one set of changes to ranks. In 5-card Short Deck, for example, the three hands listed should be ranked Flush > Full House > Straight.

## Keywords

poker, playing cards, full house, flush, straight

## Author Bios

Mark Bollman is Professor of Mathematics and chairman of the Department of Mathematics & Computer Science at Albion College in Michigan. He has taught courses in applied probability and gambling math in several undergraduate courses, and he and his students have traveled to casinos in Michigan and Nevada to compare theory and practice. Mark has published 4 books on gambling mathematics, including Basic Gambling Mathematics: The Numbers Behind The Neon.

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May 25th, 9:00 AM May 25th, 10:30 AM

When Does A Straight Beat A Full House?

Park MGM, Las Vegas, NV

In 5-card poker played with the standard 52-card deck, full houses beat flushes, which beat straights. In this paper, we look at how alternate decks with different numbers of suits or ranks lead to different orderings of these 3 hands. We find that all 6 orderings of the 3 hands are possible with decks of fewer than 200 cards and that the standard deck sits at a precarious point: varying either the 4 suits or 13 ranks often leads to a deck where the usual order is no longer correct.

Implications: For possible poker games played with nonstandard decks, changing the deck can change the game; we outline one set of changes to ranks. In 5-card Short Deck, for example, the three hands listed should be ranked Flush > Full House > Straight.