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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.
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 |
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
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