|
|
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
|
|
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%.
|