Introduction
Five to Ten 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 | 8 high | 41,964 | 0.000314 |
| Flush | 7 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 |
(Video) All about Video Poker with casino gambling expert Michael "Wizard of Odds" Shackleford
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 may 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 in Five Card Stud
The median five-card stud poker hand is ace,king,queen,jack,6.
Probabilities in Two-Player Texas Hold 'Em
For this topic, please see my separate page on probabilities in Two-Player Texas Hold 'Em.
Median Hand in Texas Hold 'Em
The following table shows the median hand in Texas Hold 'Em by the number of players. This table assumes that nobody ever folds.
(Video) Three Card Poker by the Wizard of Odds
Median Hand in Texas Hold 'Em
| Players | Median Hand |
|---|---|
| 1 | Pair -- JJAT8 |
| 2 | Two pair -- 8877J |
| 3 | Two pair -- QQ44T |
| 4 | Two pair -- KKTTJ |
| 5 | Two pair -- AAJJ6 |
| 6 | Three of a kind -- 555T7 |
| 7 | Three of a kind -- 999T8 |
| 8 | Three of a kind -- QQQT8 |
| 9 | Three of a kind -- AAAQT |
| 10 | Straight -- 65432 |
High Hand Probabilities in Texas Hold 'Em
The following two tables show the probability of the winning hand in Texas Hold 'Em for 2 to 10 players, assuming nobody ever folds. These tables were created to help me analyze Bet on Poker.
Texas Hold 'Em -- High Hand Probabilities -- 1 to 5 Players
| Hand | 1 Player | 2 Players | 3 Players | 4 Players | 5 Players |
|---|---|---|---|---|---|
| Royal Flush | 0.000032 | 0.000063 | 0.000094 | 0.000125 | 0.000155 |
| Straight Flush | 0.000279 | 0.000541 | 0.000801 | 0.001058 | 0.001312 |
| Four of a kind | 0.001680 | 0.003120 | 0.004560 | 0.005998 | 0.007437 |
| Full House | 0.025963 | 0.047027 | 0.065383 | 0.081605 | 0.096110 |
| Flush | 0.030258 | 0.052479 | 0.070330 | 0.085043 | 0.097491 |
| Straight | 0.046197 | 0.080196 | 0.107931 | 0.130825 | 0.149905 |
| Three of a kind | 0.048301 | 0.071897 | 0.092473 | 0.110374 | 0.125849 |
| Two pair | 0.234949 | 0.314178 | 0.339171 | 0.337678 | 0.323925 |
| Pair | 0.438221 | 0.371200 | 0.299665 | 0.241083 | 0.195945 |
| High card | 0.174120 | 0.059299 | 0.019591 | 0.006212 | 0.001869 |
Texas Hold 'Em -- High Hand Probabilities -- 6 to 10 Players
| Hand | 6 Players | 7 Players | 8 Players | 9 Players | 10 Players |
|---|---|---|---|---|---|
| Royal Flush | 0.000186 | 0.000217 | 0.000248 | 0.000279 | 0.000309 |
| Straight Flush | 0.001565 | 0.001814 | 0.002061 | 0.002305 | 0.002547 |
| Four of a kind | 0.008874 | 0.010311 | 0.011747 | 0.013183 | 0.014620 |
| Full House | 0.109234 | 0.121232 | 0.132294 | 0.142568 | 0.152169 |
| Flush | 0.108291 | 0.117875 | 0.126536 | 0.134473 | 0.141819 |
| Straight | 0.165917 | 0.179431 | 0.190885 | 0.200623 | 0.208920 |
| Three of a kind | 0.139101 | 0.150285 | 0.159563 | 0.167088 | 0.172997 |
| Two pair | 0.305116 | 0.284805 | 0.264703 | 0.245564 | 0.227706 |
| Pair | 0.161189 | 0.133893 | 0.111931 | 0.093911 | 0.078910 |
| High card | 0.000528 | 0.000138 | 0.000033 | 0.000007 | 0.000001 |
Chopped Pots in Texas Hold 'Em
Due to the five community cards, chops, or ties, are not unusual in Texas Hold 'Em. The following table shows the following for 2 to 10 players, assuming nobody ever folds:
- Probability of a chopped pot.
- Average winners per hand.
- Average winners per chopped pot.
Texas Hold 'Em — Chopped Pots
| Players | Prob. Chop | Average Winners per Hand | Average Winners per Chop |
|---|---|---|---|
| 2 | 0.062166 | 1.062166 | 2.000000 |
| 3 | 0.060548 | 1.074952 | 2.237896 |
| 4 | 0.066752 | 1.084148 | 2.260614 |
| 5 | 0.074879 | 1.093894 | 2.253952 |
| 6 | 0.083382 | 1.104460 | 2.252781 |
| 7 | 0.091632 | 1.115304 | 2.258336 |
| 8 | 0.099642 | 1.126432 | 2.268862 |
| 9 | 0.107048 | 1.137478 | 2.284261 |
| 10 | 0.113972 | 1.148436 | 2.302389 |
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.
(Video) All About Three Card Poker with Michael "Wizard of Odds" Shackleford
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 |
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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 |
The next table is for a seven-card stud game with one fully wild joker.
Seven Card Stud — One Joker
| Hand | Combinations | Probabilities |
|---|---|---|
| Natural Royal Flush | 4,512 | 0.000029 |
| Five of a Kind | 14,664 | 0.000095 |
| Wild Royal Flush | 21,620 | 0.000140 |
| Straight Flush | 184,832 | 0.001199 |
| Four of a Kind | 1,121,024 | 0.007273 |
| Full House | 5,997,144 | 0.038906 |
| Flush | 6,027,224 | 0.039101 |
| Straight | 11,083,940 | 0.071907 |
| Three of a kind | 13,315,300 | 0.086383 |
| Two pair | 31,433,400 | 0.203924 |
| Pair | 61,644,960 | 0.399920 |
| All other | 23,294,460 | 0.151122 |
| Total | 154,143,080 | 1.000000 |
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 |
(Video) Ten Best Bets in the Casino
The probability for a tie in a two-player game of five-card stud is 0.000344739, or 1 in 2,901.
Internal Links
- Video: Probabilities in Five Card Stud Poker
- Poker Math - How to derive the combinations for five-card stud.
- Probabilities in Two-Player Texas Hold 'Em — Lots of tables based on two-player Texas Hold 'Em.
Written by:Michael Shackleford