Swivelling Probabilities—The Winning Hand at Poker

Abstract

Poker has become a very popular card game both privately, in casinos, and on the Internet. A variant of the game known as Texas Hold’em (THm) which uses common or “community” cards is now televised widely with prize-winning tournaments and has a large following. This article examines the odds of poker hands up to the 8-card variant and considers the effect on these odds of systematically increasing the number of players at the table. It further considers the probability distribution of the “winning hand” in poker across different variants of poker, for a varying number of participants, to show that particularly for the 7-card variants with more than five players, the probability of a straight, or higher hand, winning becomes comparatively large. It concludes that as the number of players increases, the probabilities of the winning hand “swivels” around some particular hand, in that the probabilities of hands winning which are ranked below this swivel point hand decrease monotonically, whereas the probabilities of hands winning which are ranked above this swivel point increase monotonically. This increase and decrease effect on the probability of winning is muted by the introduction of a poker system, such as THm, which uses community cards.