Durango Bill's
Poker Probabilities



7 Card Poker Probabilities
for various wild card specifications
Including a “Pai Gow” (“Bug”) Joker

   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             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)


Also please see 5 card Poker probabilities
Also please see 6 card Poker probabilities
Also please see 8 card, 9 card, and 10 card Poker probabilities


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