## Wizard Endorsed Internet Casinos

## 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 |

### 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.

## 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 |

## 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.

## 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.

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!

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 * 10^{67}.

## 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 4^{5}
= 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 *
4^{2} = 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 * 6^{2} * 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 * 4^{3} = 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 4^{5}=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 |