Frequency of 7-card Poker Hands
In some popular variations of poker, a player uses the best five-card poker hand out of seven cards. The frequencies are calculated in a manner similar to that shown for 5-card hands, except additional complications arise due to the extra two cards in the 7-card poker hand. The total number of distinct 7-card hands is . It is notable that the probability of a no-pair hand is less than the probability of a one-pair or two-pair hand.
The royal flush is not included in the cumulative probability calculation because it is a type of straight flush. The Ace-high straight flush or royal flush is slightly more frequent (4324) than the lower straight flushes (4140 each) because the remaining two cards can have any value; a King-high straight flush, for example, cannot have the Ace of its suit in the hand (as that would make it ace-high instead).
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Hand Frequency Probability Cumulative Odds Royal flush 4,324 0.0032% 0.0032% 30,939 : 1 Straight flush (excl. royal flush) 37,260 0.0279% 0.0311% 3,216 : 1 Four of a kind 224,848 0.168% 0.199% 594 : 1 Full house 3,473,184 2.60% 2.80% 37.5 : 1 Flush 4,047,644 3.03% 5.82% 32.1 : 1 Straight 6,180,020 4.62% 10.4% 20.6 : 1 Three of a kind 6,461,620 4.83% 15.3% 19.7 : 1 Two pair 31,433,400 23.5% 38.8% 3.26 : 1 One pair 58,627,800 43.8% 82.6% 1.28 : 1 No pair 23,294,460 17.4% 100% 4.74 : 1 Total 133,784,560 100% 100% 0 : 1
(The frequencies given are exact; the probabilities and odds are approximate.)
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. Eliminating identical hands that ignore relative suit values leaves 6,009,159 distinct 7-card hands.
The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards because some 5-card hands are impossible with 7 cards (e.g. 7-high).
Read more about this topic: Poker Probability
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