Poker probability

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In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.

Contents

Frequency of 5-card poker hands

The following enumerates the (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement. Wild cards are not considered. The probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand by the total number of 5-card hands (the sample space, \, \begin{matrix} {52 \choose 5} = 2,598,960 \end{matrix} five-card hands). The odds are defined as the ratio (1/p) - 1 : 1, where p is the probability. Note that the cumulative column contains the probability of being dealt that hand or any of the hands ranked higher than it. (The frequencies given are exact; the probabilities and odds are approximate.)

The nCr function on most scientific calculators can be used to calculate hand frequencies; entering ​nCr​ with ​52​ and ​5​, for example, yields \begin{matrix} {52 \choose 5} = 2,598,960 \end{matrix} as above.

10 of spades Jack of spades Queen of spades King of spades Ace of spades

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