Poker/Probability

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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 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).

hand           number  Probability
straight flush    40   .000015
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
high card  1,302,540   .5012