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 acelow 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 (A2345) up to A (TJQKA) 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 fourofakinds 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 5432A to AKQJT. 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 threeofakinds 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 twopairs 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 nopair 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 nopair 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 nopair 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 fivecard 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 nopair hand is less than the probability of a onepair or twopair 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 




Additional Strip Poker Resources 




