Statistical Analysis of Texas Hold 'Em
The effect of luck (i.e., the dealing of the cards) in Texas Hold 'Em is a subject of much debate in the legal community. In the US, the laws that define "gambling" vary from state to state. Many legal jurisdictions apply a "dominance test" such that if the outcome is predominantly determined by chance, then the game is considered to be gambling. If the outcome is predominantly determined by skill, then the game is not considered gambling. This study concludes that the outcomes of 103 million observed games of Texas Hold 'Em Poker were determined by skill more often than by chance, and by a significant margin.
Separating Skill from Chance
We look at two aspects of Texas Hold 'Em that are clearly skill and chance factors. The decision to stay in the game until the end or to fold at an earlier point is strictly a matter of skill-weighing the costs and risks and potential benefits of staying in versus cutting losses and folding. The two cards each player is dealt as his "hole" cards and the "board" (the five community cards shared by all players) combine to make a 5-card hand, and the strength of that hand is strictly governed by chance.
Figure 1: Proportion of Hands Determined by Skill and Chance
The measure of how often a "showdown" occurs (where cards are revealed to determine the winner) is a measure of how often chance is allowed to shape the outcome of the game. If no showdown occurs, it means all players folded to one player who wins the game and takes all the money. When there is no showdown the cards did not determine the winner-player decisions did. When a showdown occurs, we look to see whether the player who could make the best 5-card hand actually won the game. If the player with the best 5-card hand went all the way to a showdown and won, then we can say that the outcome of that particular game was influenced by chance.
The Results
Cigital examined 103 million hands of Texas Hold 'Em poker played at PokerStars in December 2008. In the majority of cases, 75.7% of the time, the game's outcome was determined by all players folding to a single remaining player who won the pot. In the 24.3% of cases that saw a showdown, only 50.3% of those showdowns were won by the player who could make the best 5-card hand. The other roughly half of the showdowns were won by someone with an inferior 5-card hand because the player with the best 5-card hand folded prior to showdown. Figure 1 shows these three groupings of hands.