Poker probability (Texas hold 'em)
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In poker, the probability of many events can be determined by direct calculation. This article gives the probabilities and odds for many commonly occurring events in the game of Texas holdem.
For an understanding of the poker terminology and discussion, the reader may wish to read poker, poker probability, and texas holdem. For an understanding of the mathematical terms, the reader may wish to read probability, binomial coefficient, combination, sample space, and event (probability theory).
Starting hands
The probability of being dealt various starting hands can be explicitly calculated. A starting hand contains 2 out of a possible 52 cards, so the total number of possible starting hands is
<math>{52 \choose 2} = 1326<math>
Here are the probabilities of being dealt various types of starting hands.
| Hand | Frequency | Probability | Odds |
|---|---|---|---|
| Any pocket pair | 78 | .0588 | 16 : 1 |
| AA (or any specific pair) | 6 | .00452 | 220 : 1 |
| AA, KK, or QQ | 18 | .0136 | 72.7 : 1 |
| AK (or any specific no pair) | 16 | .0121 | 81.9 : 1 |
| Any 2 cards with rank at least J | 120 | .0905 | 10.1 : 1 |
| Any 2 cards with rank at least 9 | 276 | .208 | 3.80 : 1 |
| Suited cards | 312 | .235 | 3.25 : 1 |
| Connected cards | 104 | .0784 | 11.8 : 1 |
| Suited connectors | 52 | .0392 | 24.5 : 1 |
| AKs, KQs, QJs, or JTs | 16 | .0121 | 81.9 : 1 |
Derivation
Starting hands heads up
It is very useful and interesting to know how 2 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 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
- <math>{48 \choose 5} = 1712304<math>
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; there are many software programs available that will compute the odds in seconds.
For example:
The unseen cards principle states that to calculate the probability (from the point of view of a player about to act) that the next card dealt will be among a certain set, he must divide the number of cards in that set by the number of cards he has not seen, regardless of where those cards are. For example, a player playing Five-card draw who holds 5-6-7-8-K wants to discard the K hoping to draw a 4 or 9 to complete a Straight. He will calculate his probability of success as 8/47: 4 4s and 4 9s make 8 useful cards, and 52 cards minus the 5 he has already seen make 47. The fact that some of those unseen cards have already been dealt to other players is irrelevant, because he has no information about whether his desired cards are among the stub or his opponents' hands, and must act based only upon information he does have. In a game among experts, it sometimes is possible to deduce what an opponent is probably holding, and adjust your odds computation. In a stud poker or community card poker game, cards that the player has seen because they are dealt face up are subtracted from the unseen card count (and from the set of desired cards as well if they are out of play).