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, 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 as above.
| Hand | Distinct Hands | Frequency | Approx. Probability | Approx. Cumulative | Approx. Odds | Mathematical expression of absolute frequency |
|---|---|---|---|---|---|---|
| Royal flush
|
1 | 4 | 0.000154% | 0.000154% | 649,739 : 1 | |
| Straight flush (not including royal flush)
|
9 | 36 | 0.00139% | 0.00154% | 72,192.33 : 1 | |
| Four of a kind
|
156 | 624 | 0.0240% | 0.0256% | 4,164 : 1 | |
| Full house
|
156 | 3,744 | 0.144% | 0.17% | 693.17 : 1 | |
| Flush (excluding royal flush and straight flush)
|
1,277 | 5,108 | 0.197% | 0.367% | 507.8 : 1 | |
| Straight (excluding royal flush and straight flush)
|
10 | 10,200 | 0.392% | 0.76% | 253.8 : 1 | |
| Three of a kind
|
858 | 54,912 | 2.11% | 2.87% | 46.3 : 1 | |
| Two pair
|
858 | 123,552 | 4.75% | 7.62% | 20.03 : 1 | |
| One pair
|
2,860 | 1,098,240 | 42.3% | 49.9% | 1.36 : 1 | |
| No pair / High card
|
1,277 | 1,302,540 | 50.1% | 100% | .995 : 1 | |
| Total | 7,462 | 2,598,960 | 100% | --- | 0 : 1 |
The royal flush is a case of the straight flush. It can be formed 4 ways (one for each suit), giving it a probability of 0.000154% and odds of 649,739 : 1.
When ace-low straights and ace-low straight flushes are not counted, the probabilities of each are reduced: straights and straight flushes each become 9/10 as common as they otherwise would be. The 4 missed straight flushes become flushes and the 1,020 missed straights become no pair.
Note that since suits have no relative value in poker, two hands can be considered identical if one hand can be transformed into the other by swapping suits. For example, the hand 3♣ 7♣ 8♣ Q♠ A♠ is identical to 3♦ 7♦ 8♦ Q♥ A♥ because replacing all of the clubs in the first hand with diamonds and all of the spades with hearts produces the second hand. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands.
The number of distinct poker hands is even smaller. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. However, even though the hands are not identical from that perspective, they still form equivalent poker hands because each hand is an A-Q-8-7-3 high card hand. There are 7,462 distinct poker hands.
Read more about this topic: Poker Probability
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