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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
|
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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
|
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13
|
0.00294715
|
0.00000000
|
0.00058826
|
0.00000008
|
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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
|
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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
|
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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
|
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29
|
0.12391628
|
0.00000000
|
0.01954091
|
0.00025441
|
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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
|
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35
|
0.27192783
|
0.00000000
|
0.04307870
|
0.00139504
|
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36
|
0.30350759
|
0.00000000
|
0.04846353
|
0.00179362
|
|
37
|
0.33684876
|
0.00000000
|
0.05433790
|
0.00228842
|
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38
|
0.37178933
|
0.00000000
|
0.06073059
|
0.00289866
|
|
39
|
0.40812916
|
0.00000000
|
0.06767123
|
0.00364670
|
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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
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51
|
0.84324614
|
0.00000891
|
0.20560286
|
0.03774293
|
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52
|
0.86955207
|
0.00001654
|
0.22273644
|
0.04460528
|
|
53
|
0.89318170
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0.00003023
|
0.24091900
|
0.05253511
|
|
54
|
0.91406974
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0.00005441
|
0.26019252
|
0.06167165
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