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wild cellspacing=1 cellpadding=3> colspan=14>Wild Five Card Stud Probabilities wild for four-card stud with no jokers.

Probabilities in Poker

Last update: June 21, 2007

Five to Seven Card Stud

Below are the number of ways to draw each poker hand in five, six and seven card stud, and the probability of drawing them.

Five Card Stud

Hand Combinations Probability

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

0.501177394

Six Card Stud

Hand Combinations Probability

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 Probability

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

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 Probability

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 Probability

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 Probability

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 Probability

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.

Methodology for Seven Card Stud

I started to try to figure out the odds of seven card stud mathematically but it became too complicated and error prone. Instead I wrote a computer program to play out all 133,784,560 possible seven card variations and score each hand individually.

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 Decks

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

Poker Probabilities for 1 to 8 Decks

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.

Hand 2's
Four Wilds
Five of a Kind
Straight Flush
Four of a Kind
Full House
Flush
Straight
Three of a kind
Two pair
Pair
Nothing
Total

Four Card Stud

The next table is cellspacing=1 cellpadding=3> colspan=14>Wild Five Card Stud Combinations

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

624

2552

2412

2272

2132

2272

2412

2552

2416

31552

31568

31584

31600

31584

31568

31552

31600

12672

14472

14596

14720

14844

14720

14596

14472

14560

62232

57108

51984

46860

51984

57108

62232

60144

355080

356160

357240

358320

357240

356160

355080

356160

95040

1225008

1228032

1231056

1234080

1231056

1228032

1225008

1226016

799680

800700

801720

802740

801720

800700

799680

2598960

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

0.000018

0.00024

0.000982

0.000928

0.000874

0.00082

0.000874

0.000928

0.000982

0.00093

0.01214

0.012146

0.012153

0.012159

0.012153

0.012146

0.01214

0.012159

0.004876

0.005568

0.005616

0.005664

0.005712

0.005664

0.005616

0.005568

0.005602

0.023945

0.021973

0.020002

0.01803

0.020002

0.021973

0.023945

0.023142

0.136624

0.137039

0.137455

0.137871

0.137455

0.137039

0.136624

0.137039

0.036568

0.471345

0.472509

0.473673

0.474836

0.473673

0.472509

0.471345

0.471733

0.307692

0.308085

0.308477

0.30887

0.308477

0.308085

0.307692

Four Card Stud - No Jokers
Hand	Combinations	Probability
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 Probability

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 Probability

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

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.

Suggested Link: dicedealer.com has a good article on basic casino math, which goes through more slowly and patiently than I do above.

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