Probabilities

Probabilities in Poker

Below are the number of ways to draw each hand and the probability of drawing for the first draw in five-card draw and in seven-card stud.

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

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 Draw

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

Probabilities in Bingo

The following table shows the probability of forming a bingo, black out, or four corners within a specified number of calls. For example the probability of a single player forming a bingo within 25 calls is 0.06396106, or about 6.4%.

Probabilities in Bingo
Number of Calls	Bingo	Black Out	Four Corners	X
1	0.00000000	0.00000000	0.00000000	0.00000000
2	0.00000000	0.00000000	0.00000000	0.00000000
3	0.00000000	0.00000000	0.00000000	0.00000000
4	0.00000329	0.00000000	0.00000082	0.00000000
5	0.00001692	0.00000000	0.00000411	0.00000000
6	0.00005215	0.00000000	0.00001234	0.00000000
7	0.00012492	0.00000000	0.00002880	0.00000000
8	0.00025632	0.00000000	0.00005759	0.00000000
9	0.00047305	0.00000000	0.00010367	0.00000000
10	0.00080783	0.00000000	0.00017278	0.00000000
11	0.00129986	0.00000000	0.00027150	0.00000001
12	0.00199521	0.00000000	0.00040726	0.00000003
13	0.00294715	0.00000000	0.00058826	0.00000008
14	0.00421648	0.00000000	0.00082356	0.00000018
15	0.00587167	0.00000000	0.00112304	0.00000038
16	0.00798905	0.00000000	0.00149739	0.00000076
17	0.01065272	0.00000000	0.00195812	0.00000144
18	0.01395440	0.00000000	0.00251759	0.00000259
19	0.01799309	0.00000000	0.00318894	0.00000448
20	0.02287445	0.00000000	0.00398618	0.00000747
21	0.02871003	0.00000000	0.00492410	0.00001206
22	0.03561614	0.00000000	0.00601835	0.00001895
23	0.04371249	0.00000000	0.00728537	0.00002906
24	0.05312045	0.00000000	0.00874244	0.00004359
25	0.06396106	0.00000000	0.01040767	0.00006411
26	0.07635261	0.00000000	0.01229997	0.00009260
27	0.09040799	0.00000000	0.01443910	0.00013159
28	0.10623163	0.00000000	0.01684561	0.00018423
29	0.12391628	0.00000000	0.01954091	0.00025441
30	0.14353947	0.00000000	0.02254720	0.00034692
31	0.16515993	0.00000000	0.02588753	0.00046759
32	0.18881391	0.00000000	0.02958575	0.00062345
33	0.21451154	0.00000000	0.03366654	0.00082296
34	0.24223348	0.00000000	0.03815542	0.00107617
35	0.27192783	0.00000000	0.04307870	0.00139504
36	0.30350759	0.00000000	0.04846353	0.00179362
37	0.33684876	0.00000000	0.05433790	0.00228842
38	0.37178933	0.00000000	0.06073059	0.00289866
39	0.40812916	0.00000000	0.06767123	0.00364670
40	0.44563111	0.00000000	0.07519026	0.00455838
41	0.48402328	0.00000001	0.08331894	0.00566344
42	0.52300269	0.00000001	0.09208935	0.00699602
43	0.56224021	0.00000003	0.10153441	0.00859511
44	0.60138685	0.00000007	0.11168785	0.01050513
45	0.64008123	0.00000015	0.12258423	0.01277651
46	0.67795818	0.00000031	0.13425892	0.01546630
47	0.71465810	0.00000063	0.14674812	0.01863888
48	0.74983686	0.00000125	0.16008886	0.02236665
49	0.78317588	0.00000245	0.17431898	0.02673088
50	0.81439191	0.00000472	0.18947715	0.03182247
51	0.84324614	0.00000891	0.20560286	0.03774293
52	0.86955207	0.00001654	0.22273644	0.04460528
53	0.89318170	0.00003023	0.24091900	0.05253511
54	0.91406974	0.00005441	0.26019252	0.06167165
55	0.93221528	0.00009654	0.28059978	0.07216896
56	0.94768080	0.00016894	0.30218438	0.08419712
57	0.96058846	0.00029180	0.32499074	0.09794358
58	0.97111353	0.00049778	0.34906413	0.11361456
59	0.97947539	0.00083912	0.37445061	0.13143645
60	0.98592639	0.00139853	0.40119709	0.15165744
61	0.99073928	0.00230569	0.42935127	0.17454913
62	0.99419379	0.00376192	0.45896170	0.20040826
63	0.99656346	0.00607694	0.49007775	0.22955855
64	0.99810354	0.00972311	0.52274960	0.26235263
65	0.99904080	0.01541468	0.55702826	0.29917406
66	0.99956626	0.02422308	0.59296557	0.34043944
67	0.99983122	0.03774293	0.63061418	0.38660072
68	0.99994699	0.05832999	0.67002756	0.43814749
69	0.99998812	0.08943931	0.71126003	0.49560945
70	0.99999861	0.13610330	0.75436670	0.55955906
71	1.00000000	0.20560286	0.79940351	0.63061418
72	1.00000000	0.30840429	0.84642725	0.70944095
73	1.00000000	0.45945946	0.89549550	0.79675676
74	1.00000000	0.68000000	0.94666667	0.89333333
75	1.00000000	1.00000000	1.00000000	1.00000000

Dice Probability Basics
The Probabilities of Two Dice Totals

Before you play any dice game it is good to know the probability of any given total to be thrown. First lets look at the possibilities of the total of two dice. The table below shows the six possibilities for die 1 along the left column and the six possibilities for die 2 along the top column. The body of the table shows the sum of die 1 and die 2.

Two dice totals
Die 1	Die 2
Die 1	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

The colors of the body of the table illustrate the number of ways to throw each total. The probability of throwing any given total is the number of ways to throw that total divided by the total number of combinations (36). In the following table the specific number of ways to throw each total and the probability of throwing that total is shown.

Total	Number of combinations	Probability
2	1	2.78%
3	2	5.56%
4	3	8.33%
5	4	11.11%
6	5	13.89%
7	6	16.67%
8	5	13.89%
9	4	11.11%
10	3	8.33%
11	2	5.56%
12	1	2.78%
Total	36	100%

The following shows the probability of throwing each total in a chart format. As the chart shows the closer the total is to 7 the greater is the probability of it being thrown.

The Field Bet Example

Now that we understand the probability of throwing each total we can apply this information to the dice games in the casinos to calculate the house edge. For example consider the field bet in craps. This bet pays 1:1 (even money) if the next throw is a 3, 4, 9, 10, or 11, 2:1 (double the bet) on the 2, and 3:1 (triple the bet) on the 12. Note that there are 7 totals that win and only 4 that lose which might cause someone who didn't know better to think it was a good gamble. The player's return can be defined as the sum of the products of the probability of each event and the net return of that event. The following table shows each possible total, the net return, the probability of throwing that total, and the average return. The average return is the product of the net return and the probability. The player's return is the sum of the average returns.

Total	Net return	Probability	Average return
2	2	0.0278	0.0556
3	1	0.0556	0.0556
4	1	0.0833	0.0833
5	-1	0.1111	-0.1111
6	-1	0.1389	-0.1389
7	-1	0.1667	-0.1667
8	-1	0.1389	-0.1389
9	1	0.1111	0.1111
10	1	0.0833	0.0833
11	1	0.0556	0.0556
12	3	0.0278	0.0834
Total		1	-0.0278

The last row shows the player's return to be -.0278, in other words for every $1 bet the player can expect to lose 2.78 cents. The player's loss is the house's gain so the house edge is the product of -1 and the player's return, in this case 0.0278 or 2.78%.

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