Last Updated: Oct. 18, 2014

Poker Probabilities

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Five to Nine Card Stud



The following tables show the number of combinations and probability for each poker hand using the best five cards from out of 5 to 10 cards.

Five Card Stud

Hand Combinations Probabilities
Royal flush 4 0.00000154
Straight flush 36 0.00001385
Four of a kind 624 0.00024010
Full house 3,744 0.00144058
Flush 5,108 0.00196540
Straight 10,200 0.00392465
Three of a kind 54,912 0.02112845
Two pair 123,552 0.04753902
Pair 1,098,240 0.42256903
Nothing 1,302,540 0.50117739

Six Card Stud

Hand Combinations Probabilities
Royal flush 188 0.000009
Straight flush 1656 0.000081
Four of a kind 14664 0.000720
Full house 165984 0.008153
Flush 205792 0.010108
Straight 361620 0.017763
Three of a kind 732160 0.035963
Two pair 2532816 0.124411
Pair 9730740 0.477969
Nothing 6612900 0.324822
Total 20358520 1

Seven-Card Stud

Hand Combinations Probabilities
Royal flush 4,324 0.00003232
Straight flush 37,260 0.00027851
Four of a kind 224,848 0.00168067
Full house 3,473,184 0.02596102
Flush 4,047,644 0.03025494
Straight 6,180,020 0.04619382
Three of a kind 6,461,620 0.04829870
Two pair 31,433,400 0.23495536
Pair 58,627,800 0.43822546
Ace high or less 23,294,460 0.17411920
Total 133,784,560 1.00000000

The next table also shows the probability for seven-card stud, but in more detail.

Seven-Card Stud — Detailed

Hand Details Combinations Probability
Royal flush 4,324 0.000032
Straight flush K-9 4,140 0.000031
Straight flush Q-8 4,140 0.000031
Straight flush J-7 4,140 0.000031
Straight flush 10-6 4,140 0.000031
Straight flush 9-5 4,140 0.000031
Straight flush 8-4 4,140 0.000031
Straight flush 7-3 4,140 0.000031
Straight flush 6-2 4,140 0.000031
Straight flush 5-A 4,140 0.000031
Four of a kind Aces 17,296 0.000129
Four of a kind Kings 17,296 0.000129
Four of a kind Queens 17,296 0.000129
Four of a kind Jacks 17,296 0.000129
Four of a kind Tens 17,296 0.000129
Four of a kind Nines 17,296 0.000129
Four of a kind Eights 17,296 0.000129
Four of a kind Sevens 17,296 0.000129
Four of a kind Sixes 17,296 0.000129
Four of a kind Fives 17,296 0.000129
Four of a kind Fours 17,296 0.000129
Four of a kind Threes 17,296 0.000129
Four of a kind Twos 17,296 0.000129
Full house Aces up 271,392 0.002029
Full house Kings up 270,688 0.002023
Full house Queens up 269,984 0.002018
Full house Jacks up 269,280 0.002013
Full house Tens up 268,576 0.002008
Full house Nines up 267,872 0.002002
Full house Eights up 267,168 0.001997
Full house Sevens up 266,464 0.001992
Full house Sixes up 265,760 0.001986
Full house Fives up 265,056 0.001981
Full house Fours up 264,352 0.001976
Full house Threes up 263,648 0.001971
Full house Twos up 262,944 0.001965
Flush Ace high 1,584,704 0.011845
Flush King high 1,046,844 0.007825
Flush Queen high 657,772 0.004917
Flush Jack high 389,000 0.002908
Flush 10 high 212,224 0.001586
Flush 9 high 103,280 0.000772
Flush 7 high 41,964 0.000314
Flush 6 high 11,856 0.000089
Straight A-10 747,980 0.005591
Straight K-9 603,560 0.004511
Straight Q-8 603,560 0.004511
Straight J-7 603,560 0.004511
Straight 10-6 603,560 0.004511
Straight 9-5 603,560 0.004511
Straight 8-4 603,560 0.004511
Straight 7-3 603,560 0.004511
Straight 6-2 603,560 0.004511
Straight 5-A 603,560 0.004511
Three of a kind Aces 498,916 0.003729
Three of a kind Kings 498,916 0.003729
Three of a kind Queens 497,904 0.003722
Three of a kind Jacks 496,892 0.003714
Three of a kind Tens 495,880 0.003707
Three of a kind Nines 495,880 0.003707
Three of a kind Eights 495,880 0.003707
Three of a kind Sevens 495,880 0.003707
Three of a kind Sixes 495,880 0.003707
Three of a kind Fives 495,880 0.003707
Three of a kind Fours 496,892 0.003714
Three of a kind Threes 497,904 0.003722
Three of a kind Twos 498,916 0.003729
Two pair Aces up 5,042,736 0.037693
Two pair Kings up 4,575,744 0.034202
Two pair Queens up 4,110,588 0.030725
Two pair Jacks up 3,656,340 0.027330
Two pair Tens up 3,213,000 0.024016
Two pair Nines up 2,778,300 0.020767
Two pair Eights up 2,352,240 0.017582
Two pair Sevens up 1,934,820 0.014462
Two pair Sixes up 1,526,040 0.011407
Two pair Fives up 1,128,168 0.008433
Two pair Fours up 745,740 0.005574
Two pair Threes up 369,684 0.002763
High pair Aces 4,573,800 0.034188
High pair Kings 4,573,800 0.034188
High pair Queens 4,538,160 0.033921
High pair Jacks 4,502,520 0.033655
High pair Tens 4,466,880 0.033389
High pair Nines 4,472,820 0.033433
High pair Eights 4,472,820 0.033433
High pair Sevens 4,472,820 0.033433
High pair Sixes 4,472,820 0.033433
High pair Fives 4,466,880 0.033389
High pair Fours 4,502,520 0.033655
High pair Threes 4,538,160 0.033921
High pair Twos 4,573,800 0.034188
Nothing Ace high 12,944,820 0.096759
Nothing King high 6,386,940 0.047740
Nothing Queen high 2,719,500 0.020327
Nothing Jack high 963,480 0.007202
Nothing 10 high 248,640 0.001859
Nothing 9 high 31,080 0.000232
Total 133,784,560 1.000000


Eight Card Stud

Hand Combinations Probabilities
Royal flush 64,860 0.00008619
Straight flush 546,480 0.00072618
Four of a kind 2,529,262 0.00336098
Full house 45,652,128 0.06066420
Flush 50,850,320 0.06757175
Straight 67,072,620 0.08912853
Three of a kind 38,493,000 0.05115090
Two pair 257,760,900 0.34252204
Pair 236,092,500 0.31372828
Nothing 53,476,080 0.07106096
Total 752,538,150 1.00000000

Nine Card Stud

Hand Combinations Probabilities
Royal Flush 713,460 0.000194
Straight Flush 5,874,656 0.001597
Four of a kind 22,247,616 0.006047
Full House 423,908,824 0.115222
Flush 453,008,864 0.123131
Straight 509,071,920 0.138370
Three of a kind 151,728,780 0.041241
Two pair 1,442,570,040 0.392101
Pair 600,163,200 0.163129
Nonpaying hand 69,788,040 0.018969
Total 3,679,075,400 1.000000

Ten Card Stud

Hand Combinations Probabilities
Royal Flush 6,135,750 0.000388
Straight Flush 49,346,350 0.003119
Four of a kind 159,262,448 0.010067
Full House 2,971,045,612 0.187803
Flush 3,024,664,090 0.191192
Straight 2,797,153,740 0.176811
Three of a kind 372,408,960 0.023540
Two pair 5,560,398,330 0.351478
Pair 842,044,140 0.053226
Nonpaying hand 37,564,800 0.002375
Total 15,820,024,220 1.000000

Low Ball

Rules vary in low ball whether aces are high or low, and whether straights and flushes work against the player. The following tables look at two different sets of rules.

5-Card Low Ball, no Straights or Flushes

Hand Combinations Probabilities
Royal flush 0 0
Straight flush 0 0
Four of a kind 624 0.00024
Full house 3744 0.001441
Flush 0 0
Straight 0 0
Three of a kind 54912 0.021128
Two pair 123552 0.047539
Pair 1098240 0.422569
Ace high 0 0
King high 506880 0.195032
Queen high 337920 0.130021
Jack high 215040 0.082741
10 high 129024 0.049644
9 high 71680 0.02758
8 high 35840 0.01379
7 high 15360 0.00591
6 high 5120 0.00197
5 high 1024 0.000394
Total 2598960 1

5-Card Low Ball — Straights & Flushes Enforced

Hand Combinations Probabilities
Royal flush 4 0.000002
Straight flush 36 0.000014
Four of a kind 624 0.00024
Full house 3744 0.001441
Flush 5108 0.001965
Straight 10200 0.003925
Three of a kind 54912 0.021128
Two pair 123552 0.047539
Pair 1098240 0.422569
Ace high 502860 0.193485
King high 335580 0.129121
Queen high 213180 0.082025
Jack high 127500 0.049058
10 high 70380 0.02708
9 high 34680 0.013344
8 high 14280 0.005495
7 high 4080 0.00157
6 high 0 0
5 high 0 0
Total 2598960 1

Omaha

In Omaha the player many use any 2 of his own 4 cards, and any 3 of the 5 community cards, to form the best highest and lowest poker hand. Straights and flushes are not enforced in the low hand. For the low hand aces always count as low. Here are the probabilities for each hand.

Omaha High Hand

Hand Combinations Probabilities
Royal Flush 42807600 0.000092
Straight Flush 368486160 0.000795
Four of a kind 2225270496 0.0048
Full House 29424798576 0.063475
Flush 31216782384 0.067341
Straight 52289648688 0.112799
Three of a kind 40712657408 0.087825
Two pair 170775844104 0.368398
Pair 122655542152 0.264593
All other 13851662832 0.029881
Total 463563500400 1

Omaha Low Hand

Hand Combinations Probabilities
5 high 7439717760 0.016049
6 high 25832342400 0.055726
7 high 51687563904 0.111501
8 high 76415359104 0.164843
9 high 90496557312 0.195219
10 high 87800751360 0.189404
J high 68526662400 0.147826
Q high 39834609408 0.085931
K high 13835276928 0.029845
Pair or higher 1694659824 0.003656
Total 463563500400 1

Median Hand

I have been asked several times what the median hand is in both 5-card and 7-card stud. First let me review what median means. It is the mid point in a set of values. For example if I gave a test and the scores were 20%, 30%, 40%, 50%, 100%; the median would be 40%. This is not to be confused with the average, which in this example would be 48%.

The median 5-card stud poker hand is ace,king,queen,jack,6 or 7. There are 1298460 hands less than AKQJ6, 1298460 higher than AKQJ7, and 1020 equal to both of either hand.

John Lichtenberg wrote to me to add that if the ace, king, queen, 10, 2 is the hand in which a random hand from the remaining 47 cards has cloest to a 50/50 chance of being higher or lower. He adds that 766919 remaining hands in the deck are lower, 766777 are higher, and 243 are the same.

I have not determined the median 7-card stud hand but a reader (Rocke V.) sent me an e-mail claiming it is jack/jack/ace/10/8. Another reader (David Mitchell) wrote saying he agrees with the 5 and 7-card median hands and adds that the median 6-card hand is 5/5/king/10/7.

Multi-Deck Probabilities

I've been asked several times about the probabilities of each poker hand in multiple-deck games. Although I strongly feel poker based games should be played with only one deck, I will submit to the will of my readers and present the following tables. The first table shows the number of raw combinations, and the second the probability.

Poker Combinations for 1 to 8 DecksExpand

Hand Number of Decks
1 2 3 4 5 6 7 8
5 of a kind 0 728 10296 56784 201552 552552 1277640 2617888
Royal flush 4 128 972 4096 12500 31104 67228 131072
Straight flush 36 1152 8748 36864 112500 279936 605052 1179648
4 of a kind 624 87360 926640 4542720 15116400 39783744 89434800 179512320
Full house 3744 244608 2265120 10483200 33789600 87145344 193179168 383784960
Flush 5108 261840 2291436 10337408 32836500 83889648 184732940 365208576
Straight 10200 326400 2478600 10444800 31875000 79315200 171431400 334233600
3 of a kind 54912 3075072 27150552 122783232 390390000 997805952 2197787592 4345516032
2 pair 123552 5374512 44756712 197188992 617760000 1563982992 3422050632 6733089792
Pair 1098240 40909440 325250640 1401354240 4332900000 10875047040 23649465840 46319370240
Nothing 1302540 41681280 316517220 1333800960 4070437500 10128551040 21891789780 42681630720
total 2598960 91962520 721656936 3091033296 9525431552 23856384552 51801822072 101346274848


Infinite Decks

The following table shows the number of combinations if each card was dealt from a separate deck, which would be mathematically equivalent to an infinite number of decks.

Infinite Deck

Hand Permutations
Royal flush 480
Five of a kind 13,312
Straight flush 4,320
Four of a kind 798,720
Full house 1,597,440
Flush 1,470,960
Straight 1,224,000
Three of a kind 17,503,200
Two pair 26,254,800
Pair 175,032,000
Nothing 156,304,800
Total 380,204,032


Poker Probabilities for 1 to 8 DecksExpand

Hand Number of Decks
1 2 3 4 5 6 7 8
5 of a kind 0 0.00000792 0.00001427 0.00001837 0.00002116 0.00002316 0.00002466 0.00002583
Royal flush 0.00000154 0.00000139 0.00000135 0.00000133 0.00000131 0.0000013 0.0000013 0.00000129
Straight flush 0.00001385 0.00001253 0.00001212 0.00001193 0.00001181 0.00001173 0.00001168 0.00001164
4 of a kind 0.0002401 0.00094995 0.00128405 0.00146964 0.00158695 0.00166764 0.00172648 0.00177128
Full house 0.00144058 0.00265987 0.00313878 0.00339149 0.0035473 0.00365291 0.0037292 0.00378687
Flush 0.0019654 0.00284725 0.00317524 0.00334432 0.00344725 0.00351644 0.00356615 0.00360357
Straight 0.00392465 0.00354927 0.0034346 0.00337906 0.00334631 0.00332469 0.00330937 0.00329794
3 of a kind 0.02112845 0.03343832 0.03762252 0.03972239 0.04098397 0.04182553 0.04242684 0.04287791
2 pair 0.04753902 0.05844242 0.06201937 0.06379388 0.06485375 0.06555826 0.06606043 0.06643648
Pair 0.42256903 0.44484905 0.4506998 0.4533611 0.45487703 0.45585478 0.45653734 0.45704068
Nothing 0.50117739 0.45324204 0.4385979 0.4315065 0.42732316 0.42456354 0.42260656 0.42114652
total 1 1 1 1 1 1 1 1

Wild Deck Probabilities

The next two tables show the probabilities in 5-card stud with one wild card. The first table is for a partially wild card that can only be used to complete a straight, flush, straight flush, or royal flush, otherwise it must be used as an ace (same usage as in pai gow poker). The second table is for a fully wild card.

5-Card Stud with Partially Wild Joker

Hand Natural
Combinations
Wild
Combinations
Total
Combinations
Probability
five of a kind 0 1 1 0
royal flush 4 20 24 0.000008
straight flush 36 144 180 0.000063
four of a kind 624 204 828 0.000289
full house 3744 624 4368 0.001522
flush 5108 2696 7804 0.002719
straight 10200 10332 20532 0.007155
3 of a kind 54912 8448 63360 0.022079
2 pair 123552 15048 138600 0.048298
pair 1098240 116784 1215024 0.4234
nothing 1302540 116424 1418964 0.494467
total 2598960 270725 2869685 1

5-Card Stud with Fully-Wild Joker

Hand Natural
Combinations
Wild
Combinations
Total
Combinations
Probability
five of a kind 0 13 13 0.000005
royal flush 4 20 24 0.000008
straight flush 36 144 180 0.000063
four of a kind 624 2496 3120 0.001087
full house 3744 2808 6552 0.002283
flush 5108 2696 7804 0.002719
straight 10200 10332 20532 0.007155
3 of a kind 54912 82368 137280 0.047838
2 pair 123552 0 123552 0.043054
pair 1098240 169848 1268088 0.441891
nothing 1302540 0 1302540 0.453897
Total 2598960 270725 2869685 1

The next table shows the combinations and probability with two fully-wild jokers. Note that the full house and four of a kind are equal in probability.

5-Card Stud with Two Fully-Wild Jokers

Hand Natural
Combinations
One Joker
Combinations
Two Jokers
Combinations
Total
Combinations
Probability
Five of a Kind 0 26 52 78 0.000025
Royal flush 4 40 40 84 0.000027
Straight Flush 36 288 216 540 0.000171
Four of a Kind 624 4992 3744 9360 0.002960
Full House 3744 5616 0 9360 0.002960
Flush 5108 5392 888 11388 0.003601
Straight 10200 20664 3840 34704 0.010974
Three of a kind 54912 164736 13320 232968 0.073666
Two pair 123552 0 0 123552 0.039068
Pair 1098240 339696 0 1437936 0.454682
Nonpaying hand 1302540 0 0 1302540 0.411869
Total 2598960 541450 22100 3162510 1

The next table shows the number of combinations for each hand when a particular rank is wild.

Wild Five-Card Stud CombinationsExpand

Hand 2's wild 3's wild 4's wild 5's wild 6's wild 7's wild 8's wild 9's wild 10's wild J's wild Q's wild K's wild A's wild
Four Wilds 48 48 48 48 48 48 48 48 48 48 48 48 48
Five of a Kind 624 624 624 624 624 624 624 624 624 624 624 624 624
Straight Flush 2552 2412 2272 2132 2132 2132 2132 2132 2132 2272 2412 2552 2416
Four of a Kind 31552 31568 31584 31600 31600 31600 31600 31600 31600 31584 31568 31552 31600
Full House 12672 12672 12672 12672 12672 12672 12672 12672 12672 12672 12672 12672 12672
Flush 14472 14596 14720 14844 14844 14844 14844 14844 14844 14720 14596 14472 14560
Straight 62232 57108 51984 46860 46860 46860 46860 46860 46860 51984 57108 62232 60144
Three of a kind 355080 356160 357240 358320 358320 358320 358320 358320 358320 357240 356160 355080 356160
Two pair 95040 95040 95040 95040 95040 95040 95040 95040 95040 95040 95040 95040 95040
Pair 1225008 1228032 1231056 1234080 1234080 1234080 1234080 1234080 1234080 1231056 1228032 1225008 1226016
Nothing 799680 800700 801720 802740 802740 802740 802740 802740 802740 801720 800700 799680 799680
Total 2598960 2598960 2598960 2598960 2598960 2598960 2598960 2598960 2598960 2598960 2598960 2598960 2598960

Wild Five-Card Stud ProbabilitiesExpand

Hand 2's wild 3's wild 4's wild 5's wild 6's wild 7's wild 8's wild 9's wild 10's wild J's wild Q's wild K's wild A's wild
Four Wilds 0.000018 0.000018 0.000018 0.000018 0.000018 0.000018 0.000018 0.000018 0.000018 0.000018 0.000018 0.000018 0.000018
Five of a Kind 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024 0.00024
Straight Flush 0.000982 0.000928 0.000874 0.00082 0.00082 0.00082 0.00082 0.00082 0.00082 0.000874 0.000928 0.000982 0.00093
Four of a Kind 0.01214 0.012146 0.012153 0.012159 0.012159 0.012159 0.012159 0.012159 0.012159 0.012153 0.012146 0.01214 0.012159
Full House 0.004876 0.004876 0.004876 0.004876 0.004876 0.004876 0.004876 0.004876 0.004876 0.004876 0.004876 0.004876 0.004876
Flush 0.005568 0.005616 0.005664 0.005712 0.005712 0.005712 0.005712 0.005712 0.005712 0.005664 0.005616 0.005568 0.005602
Straight 0.023945 0.021973 0.020002 0.01803 0.01803 0.01803 0.01803 0.01803 0.01803 0.020002 0.021973 0.023945 0.023142
Three of a kind 0.136624 0.137039 0.137455 0.137871 0.137871 0.137871 0.137871 0.137871 0.137871 0.137455 0.137039 0.136624 0.137039
Two pair 0.036568 0.036568 0.036568 0.036568 0.036568 0.036568 0.036568 0.036568 0.036568 0.036568 0.036568 0.036568 0.036568
Pair 0.471345 0.472509 0.473673 0.474836 0.474836 0.474836 0.474836 0.474836 0.474836 0.473673 0.472509 0.471345 0.471733
Nothing 0.307692 0.308085 0.308477 0.30887 0.30887 0.30887 0.30887 0.30887 0.30887 0.308477 0.308085 0.307692 0.307692
Total 1 1 1 1 1 1 1 1 1 1 1 1 1

Four Card Stud

The next table is for four-card stud with no jokers.

Four Card Stud — No Jokers

Hand Combinations Probabilities
Four of a kind 13 0.000048
Straight Flush 44 0.000163
Flush 2816 0.010402
Straight 2772 0.010239
Three of a kind 2496 0.00922
Two pair 2808 0.010372
Pair 82368 0.30425
Nonpaying hand 177408 0.655307
Total 270725 1

The next table is for four-card stud one fully wild joker.

Four Card Stud — One Joker

Hand Combinations Probabilities
Four of a kind 65 0.000222
Straight Flush 180 0.000615
Flush 3824 0.013059
Straight 4812 0.016433
Three of a kind 6240 0.02131
Two pair 2808 0.009589
Pair 97488 0.332922
Nonpaying hand 177408 0.60585
Total 292825 1

The next table is for four-card stud with two fully-wild jokers.

Four Card Stud — Two Jokers

Hand Combinations Probabilities
Four of a kind 195 0.000617
Straight Flush 460 0.001455
Flush 5000 0.01581
Straight 7284 0.023032
Three of a kind 10488 0.033164
Two pair 2808 0.008879
Pair 112608 0.356072
Nonpaying hand 177408 0.560972
Total 316251 1

Player Must Use Every Card

The following table shows the number of combinations for 2 to 10 cards from a single 52-card deck, with no wild cards. For the purpose of this table, a royal flush, straight flush, flush, and straight must use all cards. A royal flush is defined as an ace-high straight flush. For example, with three cards, a royal flush would be suited QKA.

2 to 10 Cards — Straight, Flush, Straight Flush, and Royal Flush Must Use All CardsExpand

Hand 2 cards 3 cards 4 cards 5 cards 6 cards 7 cards 8 cards 9 cards 10 cards
Royal flush 4 4 4 4 4 4 4 4 4
Straight flush 48 44 40 36 32 28 24 20 16
Four of a kind - - 13 624 14,664 224,848 2,529,462 22,256,520 159,455,868
Full house - - - 3,744 165,984 3,473,184 45,659,328 424,213,504 2,977,017,472
Flush 260 1,096 2,816 5,108 6,828 6,832 5,120 2,836 1,124
Straight 156 720 2,772 10,200 36,828 131,040 458,724 1,572,840 5,242,860
Three of a kind - 52 2,496 54,912 732,160 6,589,440 42,172,416 196,804,608 674,758,656
Two pair - - 2,808 123,552 2,532,816 32,123,520 282,625,200 1,836,229,824 9,178,554,528
Pair 78 3,744 82,368 1,098,240 9,884,160 63,258,624 295,206,912 1,012,137,984 2,530,344,960
Garbage 780 16,440 177,408 1,302,540 6,985,044 27,977,040 83,880,960 185,857,260 294,648,732
Total 1,326 22,100 270,725 2,598,960 20,358,520 133,784,560 752,538,150 3,679,075,400 15,820,024,220

I would like to thank Miplet for confirming the table above.

Two-Player Game

The next table shows the number of combinations for a two-player game of five-card stud. This would be easy if I assumed a separate deck for each player. But, no, your faithful Wizard counted all four trillion ways two five-card hands can be drawn from a single 52-card deck. So appreciate it!

Two-Player Game of Five-Card StudExpand

Player 1 Player 2
Royal flush Straight flush Four of a kind Full house Flush Straight Three of a kind Two pair Pair Nothing
Royal flush 12 124 1376 8496 15532 28692 126008 287208 2581200 3087108
Straight flush 124 1052 12384 76464 139836 220644 1134072 2584872 23230800 27821556
Four of a kind 1376 12384 295152 1770912 1757152 3508800 23612160 53127360 425018880 448073760
Full house 8496 76464 1770912 10048896 10849392 21664800 138707712 299242944 2494103040 2766594960
Flush 15532 139836 1757152 10849392 19852668 30908664 160912216 366764616 3296192400 3947967936
Straight 28692 220644 3508800 21664800 30908664 63665904 321320400 732380400 6582060000 7890419496
Three of a kind 126008 1134072 23612160 138707712 160912216 321320400 1948607232 4280335488 36324288000 41032615080
Two pair 287208 2584872 53127360 299242944 366764616 732380400 4280335488 9339913440 80921617920 93524977080
Pair 2581200 23230800 425018880 2494103040 3296192400 6582060000 36324288000 80921617920 714035013120 840529062000
Nothing 3087108 27821556 448073760 2766594960 3947967936 7890419496 41032615080 93524977080 840529062000 1007846286084

The probability for a tie in a two-player game of five-card stud is 0.000344739, or 1 in 2,901.

Derivations for Five Card Stud

I have been asked so many times how I derived the probabilities of drawing each poker hand that I have created this section to explain the calculation. This assumes some level mathematical proficiency; anyone comfortable with high school math should be able to work through this explanation. The skills used here can be applied to a wide range of probability problems.

The Factorial Function

If you already know about the factorial function you can skip ahead. If you think 5! means to yell the number five then keep reading.

The instructions for your living room couch will probably recommend that you rearrange the cushions on a regular basis. Let's assume your couch has four cushions. How many combinations can you arrange them in? The answer is 4!, or 24. There are obviously 4 positions to put the first cushion, then there will be 3 positions left to put the second, 2 positions for the third, and only 1 for the last one, or 4*3*2*1 = 24. If you had n cushions there would be n*(n-1)*(n-2)* ... * 1 = n! ways to arrange them. Any scientific calculator should have a factorial button, usually denoted as x!, and the fact(x) function in Excel will give the factorial of x. The total number of ways to arrange 52 cards would be 52! = 8.065818 * 1067.

The Combinatorial Function

Assume you want to form a committee of 4 people out of a pool of 10 people in your office. How many different combinations of people are there to choose from? The answer is 10!/(4!*(10-4)!) = 210. The general case is if you have to form a committee of y people out of a pool of x then there are x!/(y!*(x-y)!) combinations to choose from. Why? For the example given there would be 10! = 3,628,800 ways to put the 10 people in your office in order. You could consider the first four as the committee and the other six as the lucky ones. However you don't have to establish an order of the people in the committee or those who aren't in the committee. There are 4! = 24 ways to arrange the people in the committee and 6! = 720 ways to arrange the others. By dividing 10! by the product of 4! and 6! you will divide out the order of people in an out of the committee and be left with only the number of combinations, specifically (1*2*3*4*5*6*7*8*9*10)/((1*2*3*4)*(1*2*3*4*5*6)) = 210. The combin(x,y) function in Excel will tell you the number of ways you can arrange a group of y out of x.

Now we can determine the number of possible five card hands out of a 52 card deck. The answer is combin(52,5), or 52!/(5!*47!) = 2,598,960. If you're doing this by hand because your calculator doesn't have a factorial button and you don't have a copy of Excel, then realize that all the factors of 47! cancel out those in 52! leaving (52*51*50*49*48)/(1*2*3*4*5). The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Below are the number of combinations for each hand. Just divide by 2,598,960 to get the probability.

Royal Flush

There are four different ways to draw a royal flush (one for each suit).

Straight Flush

The highest card in a straight flush can be 5,6,7,8,9,10,Jack,Queen, or King. Thus there are 9 possible high cards, and 4 possible suits, creating 9 * 4 = 36 different possible straight flushes.

Four of a Kind

There are 13 different possible ranks of the 4 of a kind. The fifth card could be anything of the remaining 48. Thus there are 13 * 48 = 624 different four of a kinds.

Full House

There are 13 different possible ranks for the three of a kind, and 12 left for the two of a kind. There are 4 ways to arrange three cards of one rank (4 different cards to leave out), and combin(4,2) = 6 ways to arrange two cards of one rank. Thus there are 13 * 12 * 4 * 6 = 3,744 ways to create a full house.

Flush

There are 4 suits to choose from and combin(13,5) = 1,287 ways to arrange five cards in the same suit. From 1,287 subtract 10 for the ten high cards that can lead a straight, resulting in a straight flush, leaving 1,277. Then multiply for 4 for the four suits, resulting in 5,108 ways to form a flush.

Straight

The highest card in a straight can be 5,6,7,8,9,10,Jack,Queen,King, or Ace. Thus there are 10 possible high cards. Each card may be of four different suits. The number of ways to arrange five cards of four different suits is 45 = 1024. Next subtract 4 from 1024 for the four ways to form a flush, resulting in a straight flush, leaving 1020. The total number of ways to form a straight is 10*1020=10,200.

Three of a Kind

There are 13 ranks to choose from for the three of a kind and 4 ways to arrange 3 cards among the four to choose from. There are combin(12,2) = 66 ways to arrange the other two ranks to choose from for the other two cards. In each of the two ranks there are four cards to choose from. Thus the number of ways to arrange a three of a kind is 13 * 4 * 66 * 42 = 54,912.

Two Pair

There are (13:2) = 78 ways to arrange the two ranks represented. In both ranks there are (4:2) = 6 ways to arrange two cards. There are 44 cards left for the fifth card. Thus there are 78 * 62 * 44 = 123,552 ways to arrange a two pair.

One Pair

There are 13 ranks to choose from for the pair and combin(4,2) = 6 ways to arrange the two cards in the pair. There are combin(12,3) = 220 ways to arrange the other three ranks of the singletons, and four cards to choose from in each rank. Thus there are 13 * 6 * 220 * 43 = 1,098,240 ways to arrange a pair.

Nothing

First find the number of ways to choose five different ranks out of 13, which is combin(13,5) = 1287. Then subtract 10 for the 10 different high cards that can lead a straight, leaving you with 1277. Each card can be of 1 of 4 suits so there are 45=1024 different ways to arrange the suits in each of the 1277 combinations. However we must subtract 4 from the 1024 for the four ways to form a flush, leaving 1020. So the final number of ways to arrange a high card hand is 1277*1020=1,302,540.

Specific High Card

For example, let's find the probability of drawing a jack-high. There must be four different cards in the hand all less than a jack, of which there are 9 to choose from. The number of ways to arrange 4 ranks out of 9 is combin(9,4) = 126. We must then subtract 1 for the 10-9-8-7 combination which would form a straight, leaving 125. From above we know there are 1020 ways to arrange the suits. Multiplying 125 by 1020 yields 127,500 which the number of ways to form a jack-high hand. For ace-high remember to subtract 2 rather than 1 from the total number of ways to arrange the ranks since A-K-Q-J-10 and 5-4-3-2-A are both valid straights.

Here is a good site that also explains how to calculate poker probabilities.

Five Card Draw — High Card Hands

Hand Combinations Probability
Ace high 502,860 0.19341583
King high 335,580 0.12912088
Queen high 213,180 0.08202512
Jack high 127,500 0.04905808
10 high 70,380 0.02708006
9 high 34,680 0.01334380
8 high 14,280 0.00549451
7 high 4,080 0.00156986
Total 1,302,540 0.501177394

Ace/King High

For the benefit of those interested in Caribbean Stud Poker I will calculate the probability of drawing ace high with a second highest card of a king. The other three cards must all be different and range in rank from queen to two. The number of ways to arrange 3 out of 11 ranks is (11:3) = 165. Subtract one for Q-J-10, which would form a straight, and you are left with 164 combinations. As above there 1020 ways to arrange the suits and avoid a flush. The final number of ways to arrange ace/king is 164*1020=167,280.

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