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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

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%.

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