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.

Frequency of 5 card poker hands

The following enumerates the frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52, without wild cards. The probability is calculated based on 2,598,960, the total number of 5 card combinations. Here, the probability is the frequency of the hand divided by the total number of 5 card hands, and the odds are defined by (1/p) − 1 : 1, where p is the probability. (The frequencies given are exact; the probabilities and odds are approximate.)

Hand	Frequency	Probability	Odds against
Royal flush	4	0.00000154 %	649,740 : 1
Straight flush	36	0.0000154 %	64,973 : 1
Four of a kind	624	0.0240 %	4,164 : 1
Full house	3,744	0.144 %	693 : 1
Flush	5,108	0.197 %	508 : 1
Straight	10,200	0.392 %	254 : 1
Three of a kind	54,912	2.11 %	46.3 : 1
Two pair	123,552	4.75 %	20.0 : 1
One pair	1,098,240	42.3 %	1.37 : 1
No pair	1,302,540	50.1 %	0.995 : 1
Total	2,598,960	100 %	1 : 1

The royal flush is also included as a straight flush above. The royal flush can be formed 4 ways (one for each suit), giving it a probability of 0.000001539077169 and odds of 649,740 : 1.

When ace-low straights and straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes become 9/10 as common as they otherwise would be.

Derivation

The following computations show how the above frequencies were determined. To understand these derivations, the reader should be familiar with the basic properties of the binomial coefficients and their interpretation as the number of ways of choosing elements from a given set. See also: sample space and event (probability theory).

• Straight flush
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