Big Two and n-card poker probabilities

TL;DR

This paper investigates how the probabilities of poker hands like straight, flush, and full house change in n-card poker, revealing that their relative likelihoods invert at certain thresholds, affecting hand rankings in larger-card games.

Contribution

It derives probability equations for various hands in n-card poker and identifies thresholds where their likelihood orderings invert, challenging traditional assumptions.

Findings

Probability of flush less than straight when n ≤ 11, more when n > 11

Probability of full house less than straight when n ≤ 19, more when n > 19

Hand ranking order does not always match occurrence probability order in larger n-card poker

Abstract

Between the poker hands of straight, flush, and full house, which hand is more common? In standard 5-card poker, the order from most common to least common is straight, flush, full house. The same order is true for 7-card poker such as Texas hold'em. However, is the same true for n-card poker for larger n? We study the probability of obtaining these various hands for n-card poker for various values of $n\geq 5$. In particular, we derive equations for the probability of flush, straight and full house and show that the probability of flush is less than a straight when $n\leq 11$, and is more than a straight when n>11. Similarly, we show that the probability of full house is less than a straight when $n\leq 19$, and is more than a straight when n>19. This means that for games such as Big Two where the ordering of 13-card hands depends on the ordering in 5-card poker, the ranking ordering does not follow the occurrence probability ordering, contrary to what intuition suggests.