No 1 in Gambling

For any given starting hand, there are 50 × 49 ÷ 2 = 1,225 hands 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 revealed on the flop to 47 × 46 ÷ 2 = 1,081 hands.) Therefore, there are

(52/2)(50/2) / 2 = 812,175

possible head-to-head match ups in Hold 'em. (The number of 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.) However, since there are only 169 distinct starting hands, there are 169 × 1,225 = 207,025 distinct head-to-head match ups.^[1]

It is useful and interesting 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 we will see a showdown. This situation occurs quite often in no limit and tournament play. Also, studying these odds helps to demonstrate the concept of hand domination, which is important in all community card games.

This problem is considerably more complicated than determining the frequency of dealt hands. To see why, note that given both hands, there are 48 remaining unseen cards. Out of these 48 cards, we can choose any 5 to make a board. Thus, there are

(48/5) = 1,712,304

possible boards that may fall. In addition to determining the precise number of boards that give a win to each player, we also must take into account boards which split the pot, and split the number of these boards between the players.

The problem is trivial for computers to solve by brute force search; there are many software programs available that will compute the odds in seconds. A somewhat less trivial exercise is an exhaustive analysis of all of the head-to-head match ups in Texas Hold 'em, which requires evaluating each possible board for each distinct head-to-head match up, or 1,712,304 × 207,025 = 354,489,735,600 (≈354 billion) results.^[1]

Notes

1. ^ ^a b  By removing reflection and applying aggressive search tree pruning, it is possible to reduce the number of unique head-to-head hand combinations from 207,205 to less than 50,000. Reflection eliminates redundant calculations by observing that given hands h₁ and h₂, if w₁ is the probability of h₁ beating h₂ in a showdown and s is the probability of h₁ splitting the pot with h₂, then the probability w₂ of h₂ beating h₁ is w₂ = 1 − (s + w₁), thus eliminating the need to evaluate h₂ against h₁. Pruning is possible, for example, by observing that Q♥ J♥ has the same chance of winning against both 8♦ 7♣ and 8♦ 7♠ (but not the same probability as against 8♥ 7♣ because sharing the heart affects the flush possibilities for each hand.)

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.