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Poker probability

This page (in progress) is intended to cover aspects of probability that are specific to poker, or that are explained in terms of poker.

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).

The following enumerates the frequency of each hand, given all permutations of 5 cards randomly drawn from a full deck of 52. The probability is calculated based on 5 card combination of 2,598,960.

```
hand           number  Probability

straight flush    36   .0000015

4-of-a-kind      624   .00024

full house     3,744   .00144

flush          5,108   .0020

straight      10,200   .0039

3-of-a-kind   54,912   .0211

two pairs    123,552   .0475

pair       1,098,240   .4226

```

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