Poker probability is a complex subject, so it is easiest to break it down in sections. In this week’s poker tournament strategy we are following up last week’s examination of starting hands in Texas Hold’em with the more complicated heads-up starting hands.
For any given starting hand, there are 1,225 hands (50 x 49/2) that an opponent can have before the flop. After the flop, the number of possible hands an opponent can have is reduced by the three community cards that are revealed on the flop. This works out to 1,081 hands (47 x 46/2). Therefore, there are 812,175 ((52/2 x 50/2) /2). The total number of match-ups is divided by the two ways that two hands can be distributed between two players to give the number of unique match-ups.
It is useful to know how two starting hands compete against each other heads-up before the flop. In other words, we assume that neither hand will fold and that there will be a showdown. Also, studying these poker probabilities helps to demonstrate the concept of hand domination, which is an important part of Texas Hold’em.
In addition to determining the precise number of boards that give a win to each player, it is valuable to take into account the boards that split the pot, and split the number of these boards between the players. These poker probabilities are trivial for computers to solve and there are many poker software programs available that will compute the odds in seconds. A somewhat less trivial exercise is an exhaustive analysis of all of the heads-up match-ups in Texas Hold’em, which requires evaluating each possible board for each distinct head-to-head match-up (or 1,712,304 x 207,025 = 354,489,735,600 results).
That’s enough numbers for one day. We’ll return to more poker probabilities next week with more poker tournament strategies.
