Durango Bill's
Poker Probabilities
8 Card, 9
Card, and 10 Card Poker Probabilities
for various wild card specifications
Including a “Pai Gow” (“Bug”) Joker
for various wild card specifications
Including a “Pai Gow” (“Bug”) Joker
Every time you think that you have finished calculating everything everyone ever wanted to know about poker, someone comes up with still a new variation. It seems that 8 card poker does exist. Thus, I might as well calculate the probabilities when you are dealt 8 cards, and then you use the best 5 to form a regular poker hand.
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 8 card deal.
While probabilities for the best 5 card hand from a deal of 8 cards (but no wild cards) could theoretically be directly calculated via combinatorics, it would be a formidable undertaking and computational errors/oversights would be highly probable. The introduction of wild cards makes calculations via combinatorics essentially impossible. 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.
8 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 64,860 0.00008619
Other straight flush 546,480 0.00072618
4 of a kind 2,529,262 0.00336098
Full House 45,652,128 0.06066420
Flush 50,850,320 0.06757175
Ace high straight 8,941,080 0.01188123
Other straights 58,131,540 0.07724730
3 of a kind 38,493,000 0.05115090
2 pairs 257,760,900 0.34252204
One pair >= Jacks 73,957,500 0.09827741
One pair <= Tens 162,135,000 0.21545087
Ace high 34,794,480 0.04623617
King high 13,719,300 0.01823070
Queen high 4,086,600 0.00543042
Jack high 817,320 0.00108608
Ten high 58,380 0.00007758
Subtotals high card only 53,476,080 0.07106096
Total = 752,538,150 1.00000000
= COMBIN(52,8)
(Interesting observation: As with 7-card hands, it is easier to evaluate to a straight than 3 of a kind.)
8 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 17,296 0.00001951
Royal straight flush 393,468 0.00044393
Other straight flush 2,699,384 0.00304560
4 of a kind 3,509,830 0.00395999
Full House 55,933,968 0.06310790
Flush 77,037,504 0.08691812
Ace high straight 17,280,024 0.01949631
Other straights 95,635,620 0.10790158
3 of a kind 43,118,280 0.04864851
2 pairs 285,316,596 0.32191051
One pair >= Jacks 77,303,700 0.08721846
One pair <= Tens 173,189,760 0.19540260
Ace high 36,205,680 0.04084932
King high 13,719,300 0.01547890
Queen high 4,086,600 0.00461074
Jack high 817,320 0.00092215
Ten high 58,380 0.00006587
Subtotals high card only 54,887,280 0.06192697
Total = 886,322,710 1.00000000
= COMBIN(53,8)
8 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 224,848 0.00025369
Royal straight flush 393,404 0.00044386
Other straight flush 2,698,824 0.00304497
4 of a kind 12,502,462 0.01410599
Full House 77,363,088 0.08728546
Flush 71,879,544 0.08109861
Ace high straight 16,719,000 0.01886333
Other straights 90,551,700 0.10216561
3 of a kind 62,838,360 0.07089783
2 pairs 257,760,900 0.29082060
One pair >= Jacks 77,779,500 0.08775528
One pair <= Tens 162,135,000 0.18292998
Ace high 34,794,480 0.03925712
King high 13,719,300 0.01547890
Queen high 4,086,600 0.00461074
Jack high 817,320 0.00092215
Ten high 58,380 0.00006587
Subtotals high card only 53,476,080 0.06033477
Total = 886,322,710 1.00000000
= COMBIN(53,8)
8 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 1,362,504 0.00130951
Royal straight flush 1,374,196 0.00132075
Other straight flush 7,882,676 0.00757610
4 of a kind 32,709,634 0.03143749
Full House 109,074,048 0.10483194
Flush 95.965,712 0.09223341
Ace high straight 25,516,440 0.02452405
Other straights 124,421,940 0.11958292
3 of a kind 87,185,160 0.08379436
2 pairs 257,760,900 0.24773606
One pair >= Jacks 81,601,500 0.07842785
One pair <= Tens 162,135,000 0.15582925
Ace high 34,794,480 0.03344125
King high 13,719,300 0.01318573
Queen high 4,086,600 0.00392766
Jack high 817,320 0.00078553
Ten high 58,380 0.00005611
Subtotals high card only 53,476,080 0.05139629
Total = 1,040,465,790 1.00000000
= COMBIN(54,8)
8 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 1,104,324 0.00146747
Royal straight flush 783,532 0.00104119
Other straight flush 6,164,321 0.00819137
4 of a kind 25,434,641 0.03379847
Full House 82,082,136 0.10907372
Flush 69,469,516 0.09231361
Ace high straight 13,662,084 0.01815467
Other straights 93,470,862 0.12420747
3 of a kind 64,732,860 0.08601937
2 pairs 185,732,484 0.24680806
One pair >= Jacks 51,211,200 0.06805130
One pair <= Tens 121,443,750 0.16137886
Ace high 24,840,690 0.03300921
King high 9,369,990 0.01245118
Queen high 2,568,720 0.00341341
Jack high 408,660 0.00054304
Ten high 58,380 0.00007758
Subtotals high card only 37,246,440 0.04949442
Total = 752,538,150 1.00000000
= COMBIN(52,8)
8 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 8,311,524 0.01104465
Royal straight flush 5,052,660 0.00671416
Other straight flush 18,702,196 0.02485216
4 of a kind 65,699,934 0.08730446
Full House 107,188,992 0.14243662
Flush 75,893,384 0.10084988
Ace high straight 32,250,420 0.04285553
Other straights 103,457,100 0.13747755
3 of a kind 66,750,840 0.08870094
2 pairs 140,762,340 0.18705011
One pair >= Jacks 40,532,160 0.05386060
One pair <= Tens 69,255,000 0.09202856
Ace high 13,719,300 0.01823070
King high 4,086,600 0.00543042
Queen high 817,320 0.00108608
Jack high 58,380 0.00007758
Ten high 0 0.00000000
Subtotals high card only 18,681,600 0.02482479
Total = 752,538,150 1.00000000
= COMBIN(52,8)
8 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 68,504,063 0.06583980
Royal straight flush 20,026,376 0.01924751
Other straight flush 70,192,060 0.06746215
4 of a kind 181,504,595 0.17444552
Full House 136,706,896 0.13139009
Flush 110,676,100 0.10637169
Ace high straight 40,179,264 0.03861661
Other straights 138,724,872 0.13332959
3 of a kind 83,265,780 0.08002741
2 pairs 98,427,984 0.09459992
One pair >= Jacks 29,002,650 0.02787468
One pair <= Tens 50,557,500 0.04859122
Ace high 9,545,130 0.00917390
King high 2,656,290 0.00255298
Queen high 467,040 0.00044888
Jack high 29,190 0.00002805
Ten high 0 0.00000000
Subtotals high card only 12,697,650 0.01220381
Total = 1,040,465,790 1.00000000
= COMBIN(54,8)
9 Card Poker Probabilities
for various wild card specifications
Including a “Pai Gow” (“Bug”) Joker
for various wild card specifications
Including a “Pai Gow” (“Bug”) Joker
While no one seems to play poker where you are dealt 9 cards for a hand, there are always new variations that are constantly invented. Thus, I might as well calculate the probabilities when you are dealt 9 cards, and then you use the best 5 to form a regular poker hand.
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 9 card deal.
While probabilities for the best 5 card hand from a deal of 9 cards (but no wild cards) could theoretically be directly calculated via combinatorics, it would be a formidable undertaking and computational errors/oversights would be highly probable. The introduction of wild cards makes calculations via combinatorics essentially impossible. 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.
9 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 713,460 0.00019392
Other straight flush 5,874,656 0.00159678
4 of a kind 22,247,616 0.00604707
Full House 423,908,824 0.11522156
Flush 453,008,864 0.12313117
Ace high straight 75,029,340 0.02039353
Other straights 434,042,580 0.11797599
3 of a kind 151,728,780 0.04124101
2 pairs 1,442,570,040 0.39210124
One pair >= Jacks 190,335,600 0.05173463
One pair <= Tens 409,827,600 0.11139418
Ace high 52,920,840 0.01438428
King high 14,758,800 0.00401155
Queen high 2,108,400 0.00057308
Subtotals high card only 69,788,040 0.01896891
Total = 3,679,075,400 1.00000000
= COMBIN(52,9)
9 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 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 194,580 0.00004391
Royal straight flush 4,344,762 0.00098040
Other straight flush 28,877,876 0.00651633
4 of a kind 31,293,954 0.00706153
Full House 527,016,160 0.11892196
Flush 678,431,768 0.15308911
Ace high straight 131,476,254 0.02966781
Other straights 642,819,624 0.14505318
3 of a kind 161,776,500 0.03650510
2 pairs 1,542,367,152 0.34803738
One pair >= Jacks 193,907,520 0.04375551
One pair <= Tens 419,102,640 0.09457112
Ace high 53,137,560 0.01199057
King high 14,758,800 0.00333034
Queen high 2,108,400 0.00047576
Subtotals high card only 70,004,760 0.01579668
Total = 4,431,613,550 1.00000000
= COMBIN(53,9)
9 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 2,529,462 0.00057078
Royal straight flush 4,341,770 0.00097973
Other straight flush 28,853,052 0.00651073
4 of a kind 108,109,392 0.02439504
Full House 697,971,070 0.15749818
Flush 595,586,144 0.13439487
Ace high straight 120,999,180 0.02730364
Other straights 575,073,540 0.12976618
3 of a kind 184,761,780 0.04169176
2 pairs 1,442,570,040 0.32551801
One pair >= Jacks 191,202,480 0.04314512
One pair <= Tens 409,827,600 0.09247819
Ace high 52,920,840 0.01194166
King high 14,758,800 0.00333034
Queen high 2,108,400 0.00047576
Subtotals high card only 69,788,040 0.01574777
Total = 4,431,613,550 1.00000000
= COMBIN(53,9)
9 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 15,346,396 0.00288578
Royal straight flush 14,994,664 0.00281964
Other straight flush 82,169,476 0.01545138
4 of a kind 261,370,848 0.04914892
Full House 972,033,316 0.18278394
Flush 752,573,900 0.14151616
Ace high straight 169,509,180 0.03187499
Other straights 717,888,660 0.13499384
3 of a kind 217,794,780 0.04095476
2 pairs 1,442,570,040 0.27126501
One pair >= Jacks 192,069,360 0.03611727
One pair <= Tens 409,827,600 0.07706516
Ace high 52,920,840 0.00995139
King high 14,758,800 0.00277529
Queen high 2,108,400 0.00039647
Subtotals high card only 69,788,040 0.01312314
Total = 5,317,936,260 1.00000000
= COMBIN(54,9)
9 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 11,886,468 0.00323083
Royal straight flush 8,196,878 0.00222797
Other straight flush 61,414,350 0.01669288
4 of a kind 193,807,128 0.05267822
Full House 695,369,282 0.18900653
Flush 516,652,504 0.14042999
Ace high straight 87,105,750 0.02367599
Other straights 511,815,880 0.13911535
3 of a kind 154,094,660 0.04188407
2 pairs 988,103,160 0.26857377
One pair >= Jacks 113,435,070 0.03083249
One pair <= Tens 290,704,050 0.07901552
Ace high 35,948,220 0.00977099
King high 9,382,380 0.00255020
Queen high 1,159,620 0.00031519
Subtotals high card only 46,490,220 0.01263639
Total = 3.679,075,400 1.00000000
= COMBIN(52,9)
9 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 81,968,568 0.02227967
Royal straight flush 46,182,736 0.01255281
Other straight flush 163,973,896 0.04456932
4 of a kind 434,473,536 0.11809313
Full House 837,782,440 0.22771549
Flush 479,668,304 0.13037741
Ace high straight 166,532,460 0.04526476
Other straights 449,838,900 0.12226955
3 of a kind 109,694,760 0.02981585
2 pairs 681,995,160 0.18537135
One pair >= Jacks 72,505,440 0.01970752
One pair <= Tens 137,592,000 0.03739853
Ace high 14,758,800 0.00401155
King high 2,108,400 0.00057308
Queen high 0 0.00000000
Subtotals high card only 16,867,200 0.00458463
Total = 3,679,075,400 1.00000000
= COMBIN(52,9)
9 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 621,443,162 0.11685796
Royal straight flush 155,126,828 0.02917049
Other straight flush 518,507,988 0.09750173
4 of a kind 1,058,322,674 0.19901003
Full House 1,001,045,352 0.18823944
Flush 588,619,836 0.11068576
Ace high straight 169,899,380 0.03194837
Other straights 495,385,390 0.09315369
3 of a kind 108,566,520 0.02041516
2 pairs 452,318,130 0.08505520
One pair >= Jacks 42,464,100 0.00798507
One pair <= Tens 95,167,800 0.01789563
Ace high 9,804,060 0.00184358
King high 1,265,040 0.00023788
Queen high 0 0.00000000
Subtotals high card only 11,069,100 0.00208147
Total = 5,317,936,260 1.00000000
= COMBIN(54,9)
10 Card Poker
Probabilities
Just in case anyone is interested in what kind of a 5-card Poker hand you can get out of a total deal of 10 cards, the computer program can also tabulate these results.
While probabilities for the best 5 card hand from a deal of 10 cards (but no wild cards) could theoretically be directly calculated via combinatorics, it would be a formidable undertaking and computational errors/oversights would be highly probable. 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.
10 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 6,135,750 0.00038785
Other straight flush 49,346,350 0.00311923
4 of a kind 159,262,448 0.01006714
Full House 2,971,045,612 0.18780285
Flush 3,024,664,090 0.19119213
Ace high straight 459,821,010 0.02906576
Other straights 2,337,332,730 0.14774521
3 of a kind 372,408,960 0.02354035
2 pairs 5,560,398,330 0.35147850
One pair >= Jacks 271,070,940 0.01713467
One pair <= Tens 570,973,200 0.03609180
Ace high 33,230,400 0.00210053
King high 4,334,400 0.00027398
Subtotals high card only 37,564,800 0.00237451
Total = 15,820,024,220 1.00000000
= COMBIN(52,10)
Also please see 5 card Poker probabilities
Also please see 6 card Poker probabilities
Also please see 7 card Poker probabilities
Return to the main Poker probabilities page
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