The Poker probability reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Poker probability

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.

Table of contents

1 Frequency of 5 card poker hands 2 Derivation 3 External links

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. 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 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).

Hand	Frequency	Probability	Odds
Straight flush	40	.0000154	64,973 : 1
Four of a kind	624	.000240	4,164 : 1
Full house	3,744	.00144	693 : 1
Flush	5,108	.00197	508 : 1
Straight	10,200	.00392	254 : 1
Three of a kind	54,912	.0211	46.3 : 1
Two pair	123,552	.0475	20.0 : 1
One pair (poker)	1,098,240	.423	1.366
No pair	1,302,540	.501	0.995 : 1
Total	2,598,960	1.00	0 : 1

Derivation

The following computations show how the above frequencies were determined.

• Straight flush -- Each straight flush is uniquely determined by its highest ranking card; and these ranks go from 5 (A-2-3-4-5) up to A (T-J-Q-K-A) in each of the 4 suits. Thus, the total number of straight flushes is

• Four of a kind -- First, we choose one of the 13 ranks for the 4 of a kind; then there are 52 − 4 = 48 cards remaining from which to choose the final card. Thus, the total number of four of a kinds is

• Full house -- First, we choose one of the 13 ranks and one of the 3 of the 4 suits for the 3 of a kind; then we choose one of the remaining 12 ranks and 2 of the 4 suits for the pair. Thus, the total number of full houses is

• Flush -- First, we choose one of four suits; then we choose 5 of the 13 possible ranks. Finally, we must subtract the 40 straight flushes, since these are ranked as straight flushes, not flushes. Thus, the total number of flushes is

• Straight -- First, we choose the highest ranking card; there are 10 of these, from 5 (A-2-3-4-5) to A (T-J-Q-K-A). Then we choose one of four suits for each of the 5 cards. Finally, we must subtract the 40 straight flushes, since these are ranked as straight flushes, not straights. Thus, the total number of straights is

• Three of a kind -- First, we choose one rank out of 13 for the 3 of a kind; then we choose 3 out of 4 suits for the 3 of a kind. Then we choose 2 distinct ranks out of the remaining 12 for the other 2 cards, as well as suits for each of those cards. Thus, the total number of three of a kinds is

• Two pair -- First, we choose 2 of the 13 ranks for the 2 pairs; then we choose 2 out of 4 suits for each of those 2 pairs. The final card can be any one of the 44 remaining cards not comprising the ranks of the 2 pairs. Thus, the total number of two pairs is

• One pair -- First, we choose one of the 13 ranks for the pair; then we choose 2 out of 4 suits for that pair. For the other 3 cards, we choose 3 ranks out of the remaining 12 and one of 4 suits for each of the 3 cards. Thus, the total number of one pairs is

• No pair -- We can choose 5 out of 13 ranks, discounting the 10 possible straights. Then we choose one of 4 suits for each of the 5 cards, discounting the 4 possible flushes. Alternatively, since no pair is any hand which does not fall into one of the above categories, we can take the total number of 5 card hands and subtract the sum of the above. Thus, the total number of no pair hands is

External links

• Poker - MathWorld - Wolfram Research

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