Texas hold’em poker is the most popular ‘community card’ variant of the game, featuring two cards, known as ‘hole cards’, dealt to each player and five more on the board. In other words, each player has the choice of seven cards from which to build the best five-card hand. Of course, a four-of-a-kind hand must include four cards of the same rank – from one of the thirteen ranks available – along with a single card, or ‘singleton’, of some other rank.
Overall, the probability of four-of-a-kind in seven cards is 0.168%, which represents odds of 594/1. In other words, a Texas hold’em player can expect to make four-of-a-kind, or ‘quads’, once every six hundred hands or so, on average. However, while four-of-a-kind is not an unbeatable hand – it ranks behind a straight flush of any description and, in the case of kings or lower, behind four-of-a-kind of higher rank – it is worth waiting for. Indeed, the probability of losing a hand with four-of-a-kind is 0.00001%, or 100,000/1 against.
If a player has paired hole cards, the probability of which is 5.88% or, in terms of odds, 16/1, the probability of ‘flopping’ four-of-a-kind is 0.245% or, in terms of odds, 400/1, although the probability of hitting four-of-a-kind by the fifth, and final, community card, known as the ‘river card’ increases to 2%, representing odds of 50/1. By contrast, a player without paired hole cards has just a 0.001%, or 1,000/1, chance of flopping four-of-a-kind.