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Casino Games Probabilities | Poker

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

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 flush 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 9-8-7-6-5 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.

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.

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/7. Just under half of poker hands (49.88%) are a pair or better. 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 Probabilites 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