Texas holdem odds are a complex subject, so it is easiest to break it down in sections. This week we are following up last week’s examination of starting hands in Texas Holdem poker with the more complicated heads-up starting hands. While this is stuff for the newer players, if poker hand rankings are a mystery to you, this will be a little over your head. For any given starting hand in Texas Holdem poker, there are 1,225 hands (50 x 49/2) that an opponent can have before the flop. Depending upon the strength of your hole cards in the pantheon of poker hand rankings, this is when many of those playing poker will decide to attack, hold their ground, or fold. After the flop, the number of possible poker 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 poker hands can be distributed between two players to give the number of unique match-ups. Seeing these three cards can move a hand up the poker hand ranking chart, and turn it into a contender. Those playing poker with strong preflop hole cards often raise. This gets rid of the drawing hands and renders a lot of the later math and possibilities irrelevant.
When playing poker it is useful to know how two starting poker hands compete against each other heads-up before the flop. In other words, we assume that neither of the poker hands will fold and that there will be a showdown. In this situation all the gamesmanship is out the window, and it simply comes down to the poker hand ranking. Also, studying these Texas Holdem odds helps to demonstrate the concept of hand domination, which is an important part of Texas Holdem poker.
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 those playing poker. These Texas Holdem odds are trivial for computers to solve and there are a lot of Texas Holdem poker software programs available that will compute the Texas Holdem odds in seconds. A somewhat less trivial exercise is an exhaustive analysis of all of the heads-up match-ups in Texas Holdem poker, 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). For this you need to know the poker hand rankings, and be fairly good at math. One of these is a given, the other is a little more difficult.
That’s enough numbers for one day on Texas Holdem odds, poker hands, and playing poker. We’ll return to more poker probabilities next week.
