Durango Bill's
Poker Probabilities
5 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
The tables below show the probabilities of being dealt various poker hands with different wild card specifications. Each poker hand consists of dealing 5 random cards.
While the results on the main Poker Probabilities page 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.
5 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 0.00000154
Other straight flush 36 0.00001385
4 of a kind 624 0.00024010
Full House 3,744 0.00144058
Flush 5,108 0.00196540
Ace high straight 1,020 0.00039246
Other straights 9,180 0.00353218
3 of a kind 54,912 0.02112845
2 pairs 123,552 0.04753902
One pair >= Jacks 337,920 0.13002124
One pair <= Tens 760,320 0.29254779
Ace high 502,860 0.19348509
King high 335,580 0.12912088
Queen high 213,180 0.08202512
Jack high 127,500 0.04905808
Ten high 70,380 0.02708006
Nine high 34,680 0.01334380
Eight high 14,280 0.00549451
Seven high 4,080 0.00156986
Subtotals high card only 1,302,540 0.50117739
Total = 2,598,960 1.00000000
= COMBIN(52,5)
5 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 0.00000035
Royal straight flush 24 0.00000836
Other straight flush 180 0.00006272
4 of a kind 828 0.00028853
Full House 4,368 0.00152212
Flush 7,804 0.00271946
Ace high straight 2,280 0.00079451
Other straights 18,252 0.00636028
3 of a kind 63,360 0.02207908
2 pairs 138,600 0.04829798
One pair >= Jacks 353,760 0.12327485
One pair <= Tens 861,264 0.30012493
Ace high 619,284 0.21580208
King high 335,580 0.11693966
Queen high 213,180 0.07428690
Jack high 127,500 0.04442996
Ten high 70,380 0.02452534
Nine high 34,680 0.01208495
Eight high 14,280 0.00497616
Seven high 4,080 0.00142176
Subtotals high card only 1,418,964 0.49446681
Total = 2,869,685 1.00000000
= COMBIN(53,5)
5 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 13 0.00000453
Royal straight flush 24 0.00000836
Other straight flush 180 0.00006272
4 of a kind 3,120 0.00108723
Full House 6,552 0.00228318
Flush 7,804 0.00271946
Ace high straight 2,280 0.00079451
Other straights 18,252 0.00636028
3 of a kind 137,280 0.04783800
2 pairs 123,552 0.04305420
One pair >= Jacks 481,308 0.16772154
One pair <= Tens 786,780 0.27416946
Ace high 502,860 0.17523178
King high 335,580 0.11693966
Queen high 213,180 0.07428690
Jack high 127,500 0.04442996
Ten high 70,380 0.02452534
Nine high 34,680 0.01208495
Eight high 14,280 0.00497616
Seven high 4,080 0.00142176
Subtotals high card only 1,302,540 0.45389651
Total = 2,869,685 1.00000000
= COMBIN(53,5)
5 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 78 0.00002466
Royal straight flush 84 0.00002656
Other straight flush 540 0.00017075
4 of a kind 9,360 0.00295967
Full House 9,360 0.00295967
Flush 11,388 0.00360094
Ace high straight 4,140 0.00130909
Other straights 30,564 0.00966448
3 of a kind 232,968 0.07366554
2 pairs 123,552 0.03906770
One pair >= Jacks 624,696 0.19753171
One pair <= Tens 813,240 0.25715018
Ace high 502,860 0.15900661
King high 335,580 0.10611192
Queen high 213,180 0.06740848
Jack high 127,500 0.04031608
Ten high 70,380 0.02225448
Nine high 34,680 0.01096597
Eight high 14,280 0.00451540
Seven high 4,080 0.00129011
Subtotals high card only 1,302,540 0.41186905
Total = 3,162,510 1.00000000
= COMBIN(54,5)
5 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 72 0.00002770
Royal straight flush 54 0.00002078
Other straight flush 480 0.00018469
4 of a kind 8,328 0.00320436
Full House 8,112 0.00312125
Flush 9,476 0.00364607
Ace high straight 2,442 0.00093961
Other straights 26,496 0.01019485
3 of a kind 199,758 0.07686074
2 pairs 102,960 0.03961585
One pair >= Jacks 470,452 0.18101548
One pair <= Tens 718,200 0.27634131
Ace high 419,220 0.16130298
King high 274,890 0.10576923
Queen high 170,850 0.06573783
Jack high 63,750 0.02452904
Ten high 70,380 0.02708006
Nine high 34,680 0.01334380
Eight high 14,280 0.00549451
Seven high 4,080 0.00156986
Subtotals high card only 1,052,130 0.40482732
Total = 2,598,960 1.00000000
= COMBIN(52,5)
5 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 672 0.00025856
Royal straight flush 484 0.00018623
Other straight flush 2,068 0.00079570
4 of a kind 31,552 0.01214024
Full House 12,672 0.00487580
Flush 14,472 0.00556838
Ace high straight 9,660 0.00371687
Other straights 52,572 0.02022809
3 of a kind 355,080 0.13662388
2 pairs 95,040 0.03656847
One pair >= Jacks 664,704 0.25575769
One pair <= Tens 560,304 0.21558777
Ace high 335,580 0.12912088
King high 213,180 0.08202512
Queen high 127,500 0.04905808
Jack high 70,380 0.02708006
Ten high 34,680 0.01334380
Nine high 14,280 0.00549451
Eight high 4,080 0.00156986
Seven high 0 0.00000000
Subtotals high card only 799,680 0.30769231
Total = 2,598,960 1.00000000
= COMBIN(52,5)
5 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 8,348 0.00263968
Royal straight flush 2,674 0.00084553
Other straight flush 11,950 0.00377864
4 of a kind 139,902 0.04423765
Full House 19,052 0.00602433
Flush 31,044 0.00981625
Ace high straight 18,318 0.00579223
Other straights 123,042 0.03890644
3 of a kind 708,240 0.22394870
2 pairs 77,880 0.02462601
One pair >= Jacks 844,032 0.26688675
One pair <= Tens 544,608 0.17220752
Ace high 274,890 0.08692146
King high 170,850 0.05402354
Queen high 99,450 0.03144654
Jack high 35,190 0.01112724
Ten high 34,680 0.01096597
Nine high 14,280 0.00451540
Eight high 4,080 0.00129011
Seven high 0 0.00000000
Subtotals high card only 633,420 0.20029028
Total = 3,162,510 1.00000000
= COMBIN(54,5)
Also please see 6 card Poker probabilities
Also please see 7 card Poker probabilities
Also please see 8 card, 9 card, and 10 card Poker probabilities
Return to the main Poker probabilities page
Return to Durango Bill’s home page
Web page generated via Sea Monkey's Composer HTML editor
within a Linux Cinnamon Mint 18 operating system.
(Goodbye Microsoft)