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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 hold 'em. 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. There are 169 distinct starting hands, corresponding to the 13 ranks for the first card multiplied by the 13 ranks for the second card. Then, as the order of the cards is not significant, the 156 combinations which aren't pairs would be divided by 2. However, since non-pair hole cards can be either suited or non-suited, they are re-multplied by 2, giving 169.

Here are the probabilities of being dealt various types of starting hands.

Probabilities during play

During play - that is, from the flop and onwards - drawing probabilities come down to a question of outs. All situations which have the same number of outs have the same probability of winning. For example, an inside straight draw (eg. 34 67 missing the 5 for a straight), and a full house draw (eg. 66KK drawing for one of the pairs to become three-of-a-kind) are equivalent. Each can be satisfied by four cards - four 5s in the first case, and the other two 6s and other two Kings in the second.

The probabilities of drawing these outs are easily calculated. At the flop there remain 47 unseen cards, so the chance is outs/47. At the turn there are 46 unseen cards so the probability is outs/46. For reference, some of the more common numbers of outs are given here. The cummulative probability of making a hand on either the turn or river can be determined as the complement of the odds of not making the hand. (In probability it is often useful to determine the odds that something won't happen and take the complement in order to determine the odds that it will happen.) The probability of not drawing an out is (47-outs)/47 on the turn and (46-outs)/46 on the river, giving a probability of drawing the out as the complement of the odds of not drawing an out on the turn and not drawing an out on the river which is calculated by the formula

${\displaystyle 1-({\frac {47-outs}{47}})({\frac {46-outs}{46}})}$

Starting hands heads up

It is very 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 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

${\displaystyle {48 \choose 5}=1,712,304}$

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; 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, 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).

External links

• Calculating Probability Odds in Texas Hold'em

• Poker gameplay and strategy