Because of this, one can use probability by outcomes to compute the probabilities of each classification of poker hand. The binomial coefficient can be used to calculate certain combinations of cards. Then, the counting principles of rule of sum and rule of product can be used to compute the frequency of each poker hand classification. Then, the probability of each poker hand classification is simply its. 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 (absolute) frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement.

1. Poker Hand Probabilities
2. Poker Hand Odds Chart
3. Poker Hand Probabilities Using Combinations

Brian Alspach

17 January 2000

Abstract:

There are a few 6-card poker games so it is worth looking at probabilitiesfor winning with certain kinds of hands. One chooses the highest ranked5-card poker hand among the 6 cards and values the hand based on the5-card hand. The types of 5-card poker hands in decreasing rank are

• straight flush
• 4-of-a-kind
• full house
• flush
• straight
• 3-of-a-kind
• two pairs
• a pair
• high card

The total number of 6-card poker hands is .

A straight flush is completely determined once the smallest card in thestraight flush is known. There are 40 cards eligible to be the smallestcard in a straight flush. If the smallest card in the straight flush isan ace, then the sixth card may be any of 47 cards. If the smallest cardin the straight flush is any of the other 36 eligible beginning cards,then the sixth card may be any of 46 cards because we cannot use the nextsmallest card in the same suit as the straight flush. Hence, there are6-card hands containing straight flushes.

In forming a 4-of-a-kind hand, there are 13 choices for the rank ofthe quads, 1 choice for the 4 cards of the given rank, and choices for the remaining 2 cards. This implies there are 4-of-a-kind hands.

There are 2 ways to get a full house and we count them separately. Oneway of obtaining a full house is for the 6-card hand to contain 2 setsof triples. There are ways to choose the 2 ranksand 4 ways to choose each of the triples. This gives us full houses of this type. The other way of getting a full houseis for the 6-card hand to contain a triple, a pair and a remaining card ofa third rank. There are 13 choices for the rank of the triple, 12 choicesfor the rank of the pair, and 44 choices for the singleton card. Thereare 4 ways of choosing the triple of a given rank and 6 ways to choose thepair of the other rank. This produces full houses of the latter type. Adding the two numbersyields 165,984 full houses.

To count the number of flushes, we obtain 6-card hands formed from cards in the same suit. Altogether, thereare flushes with 6 cards in the same suit.There are choices for 5 cards in the same suit.There are then 39 choices for a sixth card from a different suit. Thus,there are flushes in 6-card hands, whereprecisely 5 cards are in the same suit. Combining the two gives us207,636 6-card hands containing flushes. Of these, 1,844 are straightflushes whose removal leaves 205,792 flushes.

Let's determine how many sets of 6 distinct ranks correspond to straights.One possible form is ,where x can be any of9 ranks. The other possible form is ,where yis neither x-1 nor x+5. When x is ace or 10, then there are 7choices for y. When x is between 2 and 9, inclusive, there are 6choices for y. This implies there are sets of 6 distinct ranks corresponding to straights. There are then 4choices for each card of the given ranks except we must remove thosechoices producing flushes. There are 4 choices giving all 6 cards inthe same suit. If 5 are in the same suit, there are choices of which 5 ranks will be in the same suit, 4 choicesfor the suit of the 5 cards, and 3 choices for the suit of the remainingcard. So there are choices which give aflush. This means there are 4⁶ - 76 = 4,020 choices not producing aflush. Hence, there are straights of thisform.

We also can have a set of 5 distinct ranks producing a straight whichmeans the corresponding 6-card hand must contain a pair as well. Thereare 10 sets of ranks of the form .There are 5choices of the rank to be paired, 6 choices for the pair, and 4 choicesfor each of the other 4 cards except not all 4 cards can be chosen inthe same suit as either of the cards in the pair. This means there are4⁴ - 2 = 254 choices for the 4 cards. We then have straights of this form. Altogether there are361,620 straights.

In forming a 3-of-a-kind hand, there must be a triple and 3 other cardsall of distinct ranks different from the rank of the triple. There are13 choices for the rank of the triple, and there are choices for the ranks of the other 3 cards. There are 4 choicesfor the triple of the given rank and there are 4 choices for each of thecards of the remaining 3 ranks. Altogether, we have 3-of-a-kind hands.

Next we consider two pairs hands. Such a hand may contain either threepairs, or two pairs plus two remaining cards of distinct ranks. Weevaluate these 2 types of hands separately. If the hand has threepairs, there are ways to choose the ranks ofthe pairs and 6 ways to choose each of the pairs. This produces6-card hands with three pairs.

For the other kind of two pairs hand, there are choices for the two ranks of the pairs and there are ways to choose the ranks of the 2 singletons. There are 6 choicesfor each of the pairs, and there are 4 choices for each of the remaining2 cards. This produces hands of two pairs of the second type. Adding the two gives 2,532,8166-card hands with two pairs.

Now we count the number of hands with a pair. Such a hand must have5 distinct ranks. There are possible setsof 5 ranks. We must remove sets of the form because these correspond to straights. There are 10 such sets leaving1,277 sets of ranks corresponding to a hand with one pair. Given sucha set, there are 5 choices for the rank of the pair, and 6 choices fora pair of the chosen rank. There are 4 choices for each of the remaining4 cards except we cannot choose all 4 to be in the same suit as eitherof the cards forming the pair. Hence, there are 4⁴-2 = 254 choicesfor the remaining 4 cards. This gives us hands with a pair.

We could determine the number of high card hands by removing the handswhich have already been counted in one of the previous categories.Instead, let us count them independently and see if the numbers sumto 20,358,520 which will serve as a check on our arithmetic.

A high card hand has 6 distinct ranks, but does not include straights.So we must eliminate sets of ranks which have 5 consecutive ranks. Indetermining the number of straights above, we derived that there are 71sets of 6 distinct ranks which give straights. There are sets of 6 distinct ranks. Removing the 71 sets correspondingto straights, leaves 1,645 sets of distinct ranks which do not producestraights. There are 4 choices for each of the 6 cards in a given setproducing 4⁶ = 4,096 ways of choosing cards for a given set of ranks.However, some of the choices produce flushes and we must remove them.Clearly there are 4 ways of choosing the 6 cards all in the same suitwhich is one way of getting a flush. There 6 ways of choosing 5 of theranks and 4 choices for the suit of these 5 ranks, and 3 choices for thesuit of the remaining card. This gives us choices of suits which produce a flush with 5 cards in the same suit.We remove these 76 choices which produce flushes giving us 4,020 choicesfor the 6 cards which do not produce a flush. Multiplying 4,020 by 1,645gives 6,612,900 high card hands.

If we sum the preceding numbers, we obtain 20,358,520 and we can be confidentthe numbers are correct.

Here is a table summarizing the number of 6-card poker hands. Theprobability is the probability of having the hand dealt to you whendealt 6 cards.

hand	number	Probability
straight flush	1,844	.000091
4-of-a-kind	14,664	.00072
full house	165,984	.00815
flush	205,792	.0101
straight	361,620	.0178
3-of-a-kind	732,160	.036
two pairs	2,532,816	.1244
pair	9,730,740	.478
high card	6,612,900	.325

You will observe that you are less likely to be dealt a hand withno pair (or better) than to be dealt a hand with one pair. Thishas caused some people to query the ranking of these two hands.In fact, if you were ranking 6-card hands based on 6 cards, theorder of the last 2 would switch. However, you are basing the rankingon 5 cards so that if you were to rank a high card hand higher than a handwith a single pair, people would choose to ignore the pair in a6-card hand with a single pair and call it a high card hand. Thiswould have the effect of creating the following distortion. Thereare 16,343,640 6-card hands containing 5 cards which are high cardhands. Of these 16,343,640 hands, 9,730,740 also contain 5-cardhands which have a pair. Thus, the latter hands are more specialand should be ranked higher (as they indeed are) but would not beunder the scheme being discussed in this paragraph.

The tables below show the probabilities of being dealt various poker hands with different wild card specifications. Each Poker hand consists of selecting the 5 best cards from a random 7 card deal.
While probabilities for the best 5 card hand from a deal of 7 cards (but no wild cards) can be calculated via direct combinatorics, the introduction of wild cards greatly complicates the combinatoric calculations. Thus, to produce the results shown here, the author wrote a computer program that would generate all possible poker hands. Each of these poker hands was evaluated for matched ranks (pairs, 3 of a kind, etc.), straights, and flushes. Wild cards introduce multiple evaluations for a given hand, and the best standard evaluation for any given hand is used in the tables.
Data from this page may be freely used provided it includes an acknowledgement to the author.
7 card poker probabilities if there are no wild cards
(Computer program and data by Bill Butler)
Poker Hand Nbr. of Hands Probability
----------------------------------------------------
5 of a kind 0 0.00000000
Royal straight flush 4,324 0.00003232
Other straight flush 37,260 0.00027851
4 of a kind 224,848 0.00168067
Full House 3,473,184 0.02596102
Flush 4,047,644 0.03025494
Ace high straight 747,980 0.00559093
Other straights 5,432,040 0.04060289
3 of a kind 6,461,620 0.04829870
2 pairs 31,433,400 0.23495536
One pair >= Jacks 18,188,280 0.13595201
One pair <= Tens 40,439,520 0.30227345
Ace high 12,944,820 0.09675870
King high 6,386,940 0.04774049
Queen high 2,719,500 0.02032746
Jack high 963,480 0.00720173
Ten high 248,640 0.00185851
Nine high 31,080 0.00023231
Subtotals high card only 23,294,460 0.17411920
Total = 133,784,560 1.00000000
= COMBIN(52,7)
(Interesting observation: If a hand evaluates to just one pair, it is not distributed 4/13 “Jacks or better”. If you have a single middle-sized pair, you have a slightly increased chance of also having a straight which evaluates to a better hand. Thus a middle-sized pair occurs slightly less often than a high (Jacks or better) or a low (5’s or lower) pair.)
7 card poker probabilities if one “Pai Gow” (“Bug”) Joker is added to the deck
A “Pai Gow” (“Bug”) Joker is partially wild. If you are using it to complete a straight and/or a flush, it is an ordinary wild card. If you are using it for pairs, 3-of-a-kind, etc., it is forced to be an Ace.
(Computer program and data by Bill Butler)

Poker Hand Probabilities

Poker Hand Nbr. of Hands Probability
----------------------------------------------------
5 Aces 1,128 0.00000732
Royal straight flush 26,132 0.00016953
Other straight flush 184,832 0.00119909
4 of a kind 307,472 0.00199472
Full House 4,188,528 0.02717299
Flush 6,172,088 0.04004129
Ace high straight 1,554,156 0.01008255
Other straights 9,681,872 0.06281094
3 of a kind 7,470,676 0.04846585
2 pairs 35,553,816 0.23065464
One pair >= Jacks 19,273,104 0.12503386
One pair <= Tens 44,948,856 0.29160476
Ace high 14,430,780 0.09361938
King high 6,386,940 0.04143514
Queen high 2,719,500 0.01764270
Jack high 963,480 0.00625056
Ten high 248,640 0.00161305
Nine high 31,080 0.00020163
Subtotals high card only 24,780,420 0.16076246
Total = 154,143,080 1.00000000
= COMBIN(53,7)
7 card poker probabilities if one ordinary Joker is added to the deck
(Computer program and data by Bill Butler)
Poker Hand Nbr. of Hands Probability
----------------------------------------------------
5 of a kind 14,664 0.00009513
Royal straight flush 26,132 0.00016953
Other straight flush 184,832 0.00119909
4 of a kind 1,121,024 0.00727262
Full House 5,997,144 0.03890635
Flush 6,027,224 0.03910149
Ace high straight 1,543,460 0.01001316
Other straights 9,540,480 0.06189366
3 of a kind 13,315,300 0.08638273
2 pairs 31,433,400 0.20392352
One pair >= Jacks 21,170,640 0.13734408
One pair <= Tens 40,474,320 0.26257630
Ace high 12,944,820 0.08397925
King high 6,386,940 0.04143514
Queen high 2,719,500 0.01764270
Jack high 963,480 0.00625056
Ten high 248,640 0.00161305
Nine high 31,080 0.00020163
Subtotals high card only 23,294,460 0.15112232
Total = 154,143,080 1.00000000
= COMBIN(53,7)
7 card poker probabilities if two Jokers are added to the deck
(Computer program and data by Bill Butler)
Poker Hand Nbr. of Hands Probability
----------------------------------------------------
5 of a kind 88,608 0.00050033
Royal straight flush 91,764 0.00051815
Other straight flush 548,196 0.00309539
4 of a kind 3,134,544 0.01769923
Full House 8,521,104 0.04811449
Flush 8,397,324 0.04741557
Ace high straight 2,531,540 0.01429436
Other straights 14,181,120 0.08007383
3 of a kind 20,216,380 0.11415198
2 pairs 31,433,400 0.17748899
One pair >= Jacks 24,153,000 0.13638014
One pair <= Tens 40,509,120 0.22873513
Ace high 12,944,820 0.07309305
King high 6,386,940 0.03606392
Queen high 2,719,500 0.01535568
Jack high 963,480 0.00544030
Ten high 248,640 0.00140395
Nine high 31,080 0.00017549
Subtotals high card only 23,294,460 0.13153239
Total = 177,100,560 1.00000000
= COMBIN(54,7)
7 card poker probabilities with One-eyed Jacks wild
(Computer program and data by Bill Butler)
Poker Hand Nbr. of Hands Probability
----------------------------------------------------
5 of a kind 75,072 0.00056114
Royal straight flush 54,508 0.00040743
Other straight flush 447,946 0.00334826
4 of a kind 2,552,718 0.01908081
Full House 6,733,344 0.05032975
Flush 6,388,172 0.04774970
Ace high straight 1,404,464 0.01049795
Other straights 11,201,130 0.08372513
3 of a kind 15,758,140 0.11778743
2 pairs 23,810,436 0.17797596
One pair >= Jacks 16,255,890 0.12150797
One pair <= Tens 32,047,590 0.23954625
Ace high 9,743,580 0.07283038
King high 4,662,000 0.03484707
Queen high 1,888,110 0.01411306
Jack high 481,740 0.00360086
Ten high 248,640 0.00185851
Nine high 31,080 0.00023231
Subtotals high card only 17,055,150 0.12748220
Total = 133,784,560 1.00000000
= COMBIN(52,7)
7 card poker probabilities with Deuces (2’s) wild
(Computer program and data by Bill Butler)
Poker Hand Nbr. of Hands Probability
----------------------------------------------------
5 of a kind 609,760 0.00455778
Royal straight flush 399,484 0.00298602
Other straight flush 1,552,732 0.01160621
4 of a kind 7,504,920 0.05609706
Full House 9,421,824 0.07042535
Flush 7,993,600 0.05974979
Ace high straight 4,033,160 0.03014668
Other straights 15,355,640 0.11477887
3 of a kind 20,151,920 0.15062964
2 pairs 19,491,840 0.14569574
One pair >= Jacks 16,211,160 0.12117362
One pair <= Tens 20,708,880 0.15479275
Ace high 6,386,940 0.04774049
King high 2,719,500 0.02032746
Queen high 963,480 0.00720173
Jack high 248,640 0.00185851
Ten high 31,080 0.00023231
Nine high 0 0.00000000
Subtotals high card only 10,349,640 0.07736050
Total = 133,784,560 1.00000000
= COMBIN(52,7)
7 card poker probabilities with 2 Jokers,
One-eyed Jacks, and Deuces (2’s) wild
(8 out of 54 cards are wild)
(Computer program and data by Bill Butler)
Poker Hand Nbr. of Hands Probability
----------------------------------------------------
5 of a kind 5,496,072 0.03103362
Royal straight flush 1,821,704 0.01028627
Other straight flush 6,959,976 0.03929957
4 of a kind 23,628,576 0.13341898
Full House 12,751,424 0.07200104
Flush 13,497,668 0.07621471
Ace high straight 6,037,238 0.03408932
Other straights 25,527,008 0.14413849
3 of a kind 28,206,968 0.15927091
2 pairs 14,381,496 0.08120525
One pair >= Jacks 15,378,900 0.08683711
One pair <= Tens 16,024,260 0.09048114
Ace high 4,693,080 0.02649952
King high 1,911,420 0.01079285
Queen high 629,370 0.00355374
Jack high 124,320 0.00070197
Ten high 31,080 0.00017549
Nine high 0 0.00000000
Subtotals high card only 7,389,270 0.04172358
Total = 177,100,560 1.00000000
= COMBIN(54,7)
Alsoplease see 5 card Poker probabilities
Alsoplease see 6 card Poker probabilities
Alsoplease see 8 card, 9 card, and 10 card Poker probabilities
Return to the main Poker probabilities page
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Poker Hand Odds Chart

Poker Hand Probabilities Using Combinations

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