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Strip Poker - Hand 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.

Strip Poker - 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
Straight flush 40 0.00154 % 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 % 0 : 1

The royal flush is included as a straight flush above. By itself, 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).

Strip Poker - Hand Probability - 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:

{4 \choose 1}{10 \choose 1} = 40

Strip Poker - Hand Probability - Four of a kind

Any one of the thirteen ranks can form the four of a kind, leaving 52 - 4 = 48 possibilities for the final card. Thus, the total number of four-of-a-kinds is:

{13 \choose 1}{48 \choose 1} = 624

Strip Poker - Hand Probability - Full house

The full house comprises a triple (three of a kind) and a pair. The triple can be any one of the thirteen ranks, and any three of the four suits. The pair can be any one of the remaining twelve ranks, and any two of the four suits. Thus, the total number of full houses is:

{13 \choose 1}{4 \choose 3}{12 \choose 1}{4 \choose 2} = 3,744

Strip Poker - Hand Probability - Flush

The flush contains any five of the thirteen ranks, all of which belong to one of the four suits, minus the 40 straight flushes. Thus, the total number of flushes is:

{4 \choose 1}{13 \choose 5} - 40 = 5,108

Strip Poker - Hand Probability - Straight

Strip Poker - Hand Probability - Straight — The straight consists of any one of the ten possible sequences of five consecutive cards, from 5-4-3-2-A to A-K-Q-J-T. Each of these five cards can have any one of the four suits. Finally, as with the flush, the 40 straight flushes must be excluded, giving:

{10 \choose 1}{4 \choose 1}^5 - 40 = 10,200

Strip Poker - Hand Probability - Three of a kind

Any of the thirteen ranks can form the three of a kind, which can contain any three of the four suits. The other cards can have any two of the remaining twelve ranks, and each can have any one of the four suits. Thus, the total number of three-of-a-kinds is:

{13 \choose 1}{4 \choose 3}{12 \choose 2}{4 \choose 1}^2 = 54,912

Strip Poker - Hand Probability - Two pair

The pairs can have any two of the thirteen ranks, and each pair can have two of the four suits. The final card can have any one of the eleven remaining ranks, and any suit. Thus, the total number of two-pairs is:

{13 \choose 2}{4 \choose 2}^2{11 \choose 1}{4 \choose 1} = 123,552

Strip Poker - Hand Probability - Pair

The pair can have any one of the thirteen ranks, and any two of the four suits. The remaining three cards can have any three of the remaining twelve ranks, and each can have any of the four suits. Thus, the total number of pair hands is:

{13 \choose 1}{4 \choose 2}{12 \choose 3}{4 \choose 1}^3 = 1,098,240

Strip Poker - Hand Probability - No pair

A no-pair hand contains five of the thirteen ranks, discounting the ten possible straights, and each card can have any of the four suits, discounting the four possible flushes. Alternatively, a no-pair hand is any hand that does not fall into one of the above categories; that is, any way to choose five out of 52 cards, discounting all of the above hands. Thus, the total number of no-pair hands is:

{13 \choose 5} - 10\right](4^5 - 4) = {52 \choose 5} - 1,296,420 = 1,302,540

Strip Poker - 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, probabilities, and odds are calculated as above; however the total numbers are greater since there are 133,784,560 (over 50 times more) 7 card combinations. 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 frequencies given are exact; the probabilities and odds are approximate.)

Hand Frequency Probability Odds
Straight flush 41,584 0.03108 % 3,216 : 1
Four of a kind 224,848 0.1681 % 594 : 1
Full house 3,473,184 2.60 % 37.5 : 1
Flush 4,047,644 3.03 % 32.1 : 1
Straight 6,180,020 4.62 % 20.6 : 1
Three of a kind 6,461,620 4.83 % 19.7 : 1
Two pair 31,433,400 23.5 % 3.26 : 1
One pair 58,627,800 43.8 % 1.28 : 1
No pair 23,294,460 17.4 % 4.74 : 1
Total 133,784,560 100 % 0 : 1
 
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