|
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
|
|
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%.
|
