3-reel slot machines

1. Case A the same number of stops and symbol distribution on the reels

         1.3  Event   A specific symbol exactly twice

            The probability of  is given in the following table, where t is the number of stops of a reel and c is the distribution of that symbol on a reel. The values of t are listed in increments of 2, ranging from 20 to 200, and the values of c range from 1 to 7.

                Table of values for the probability of a specific symbol occurring twice on a payline 

c

1

2

3

4

5

6

7

t

20

0.007125

0.027

0.057375

0.096

0.140625

0.189

0.238875

22

0.0059166

0.02253944

0.04817806

0.081142

0.1197408

0.162284

0.20708114

24

0.00499132

0.01909722

0.04101563

0.06944444

0.1030816

0.140625

0.18077257

26

0.00426718

0.01638598

0.03533227

0.06008193

0.08961083

0.12289486

0.15890988

28

0.00368987

0.01421283

0.03074891

0.05247813

0.07858054

0.10823615

0.140625

30

0.00322222

0.01244444

0.027

0.04622222

0.06944444

0.096

0.12522222

32

0.00283813

0.01098633

0.02389526

0.04101563

0.0617981

0.08569336

0.1121521

34

0.00251883

0.00977

0.02129554

0.03663749

0.05533788

0.07693873

0.10098209

36

0.00225051

0.00874486

0.01909722

0.03292181

0.04983282

0.06944444

0.09137088

38

0.00202289

0.00787287

0.0172219

0.02974194

0.04510497

0.06298294

0.08304782

40

0.00182813

0.007125

0.01560938

0.027

0.04101563

0.057375

0.07579688

42

0.00166019

0.00647878

0.01421283

0.02461937

0.03745546

0.05247813

0.06944444

44

0.00151437

0.0059166

0.0129954

0.02253944

0.03433743

0.04817806

0.06385002

46

0.00138695

0.00542451

0.01192776

0.02071176

0.0315916

0.04438235

0.05889907

48

0.00127496

0.00499132

0.01098633

0.01909722

0.02916124

0.04101563

0.05449761

50

0.001176

0.004608

0.010152

0.017664

0.027

0.038016

0.050568

52

0.00108813

0.00426718

0.00940914

0.01638598

0.0250697

0.03533227

0.04704569

54

0.00100975

0.00396281

0.00874486

0.01524158

0.02333867

0.03292181

0.0438767

56

0.00093955

0.00368987

0.00814846

0.01421283

0.02178048

0.03074891

0.04101563

58

0.00087642

0.00344418

0.00761101

0.01328468

0.02037291

0.02878347

0.03842408

60

0.00081944

0.00322222

0.007125

0.01244444

0.01909722

0.027

0.03606944

62

0.00076785

0.00302105

0.00668407

0.01168138

0.01793746

0.02537679

0.03392384

64

0.00072098

0.00283813

0.00628281

0.01098633

0.01688004

0.02389526

0.03196335

66

0.00067827

0.00267134

0.0059166

0.01035145

0.01591326

0.02253944

0.03016738

68

0.00063925

0.00251883

0.00558149

0.00977

0.0150271

0.02129554

0.02851809

70

0.0006035

0.00237901

0.00527405

0.00923615

0.01421283

0.0201516

0.027

72

0.00057067

0.00225051

0.00499132

0.00874486

0.0134629

0.01909722

0.0255996

74

0.00054044

0.00213215

0.00473072

0.00829171

0.01277071

0.01812331

0.02430508

76

0.00051256

0.00202289

0.00448999

0.00787287

0.0121305

0.0172219

0.02310605

78

0.00048677

0.00192181

0.00426718

0.00748495

0.0115372

0.01638598

0.02199337

80

0.00046289

0.00182813

0.00406055

0.007125

0.01098633

0.01560938

0.02095898

82

0.00044072

0.00174112

0.00386856

0.00679038

0.01047395

0.01488661

0.01999572

84

0.00042011

0.00166019

0.00368987

0.00647878

0.00999656

0.01421283

0.01909722

86

0.00040091

0.00158477

0.00352327

0.00618813

0.00955105

0.01358371

0.01825783

88

0.00038299

0.00151437

0.00336771

0.0059166

0.00913464

0.0129954

0.01747247

90

0.00036626

0.00144856

0.00322222

0.00566255

0.00874486

0.01244444

0.01673663

92

0.00035059

0.00138695

0.00308596

0.00542451

0.00837948

0.01192776

0.01604622

94

0.00033591

0.00132919

0.00295816

0.00520116

0.00803651

0.01144255

0.0153976

96

0.00032213

0.00127496

0.00283813

0.00499132

0.00771417

0.01098633

0.01478746

98

0.00030918

0.00122398

0.00272527

0.00479392

0.00741082

0.01055683

0.01421283

100

0.000297

0.001176

0.002619

0.004608

0.007125

0.010152

0.013671

102

0.00028552

0.00113079

0.00251883

0.00443268

0.00685539

0.00977

0.01315953

104

0.0002747

0.00108813

0.00242429

0.00426718

0.0066008

0.00940914

0.0126762

106

0.00026448

0.00104784

0.00233498

0.00411078

0.00636012

0.00906789

0.01221898

108

0.00025482

0.00100975

0.00225051

0.00396281

0.00613235

0.00874486

0.01178603

110

0.00024568

0.0009737

0.00217055

0.00382269

0.0059166

0.00843877

0.01137566

112

0.00023702

0.00093955

0.00209477

0.00368987

0.00571204

0.00814846

0.01098633

114

0.00022882

0.00090716

0.00202289

0.00356385

0.00551789

0.00787287

0.01061663

116

0.00022103

0.00087642

0.00195465

0.00344418

0.00533348

0.00761101

0.01026526

118

0.00021363

0.00084721

0.0018898

0.00333043

0.00515815

0.007362

0.00993103

120

0.0002066

0.00081944

0.00182813

0.00322222

0.00499132

0.007125

0.00961285

122

0.00019991

0.00079302

0.00176942

0.0031192

0.00483245

0.00689926

0.0093097

124

0.00019354

0.00076785

0.0017135

0.00302105

0.00468105

0.00668407

0.00902066

126

0.00018746

0.00074386

0.00166019

0.00292745

0.00453665

0.00647878

0.00874486

128

0.00018167

0.00072098

0.00160933

0.00283813

0.00439882

0.00628281

0.0084815

130

0.00017615

0.00069914

0.00156076

0.00275284

0.00426718

0.00609558

0.00822986

132

0.00017087

0.00067827

0.00151437

0.00267134

0.00414136

0.0059166

0.00798924

134

0.00016583

0.00065833

0.00147001

0.0025934

0.00402102

0.00574539

0.00775902

136

0.000161

0.00063925

0.00142757

0.00251883

0.00390585

0.00558149

0.00753859

138

0.00015639

0.00062099

0.00138695

0.00244742

0.00379556

0.00542451

0.00732742

140

0.00015197

0.0006035

0.00134803

0.00237901

0.00368987

0.00527405

0.007125

142

0.00014773

0.00058674

0.00131073

0.00231342

0.00358853

0.00512977

0.00693084

144

0.00014367

0.00057067

0.00127496

0.00225051

0.00349131

0.00499132

0.00674451

148

0.00013604

0.00054044

0.00120767

0.00213215

0.00330836

0.00473072

0.00639369

150

0.00013244

0.00052622

0.001176

0.00207644

0.00322222

0.004608

0.00622844

152

0.00012899

0.00051256

0.00114556

0.00202289

0.00313941

0.00448999

0.00606952

154

0.00012568

0.00049942

0.00111629

0.00197138

0.00305975

0.00437646

0.0059166

156

0.00012248

0.00048677

0.00108813

0.00192181

0.00298308

0.00426718

0.00576939

158

0.00011941

0.00047461

0.00106102

0.00187409

0.00290925

0.00416194

0.0056276

160

0.00011646

0.00046289

0.00103491

0.00182813

0.00283813

0.00406055

0.00549097

162

0.00011361

0.0004516

0.00100975

0.00178383

0.00276959

0.00396281

0.00535925

164

0.00011086

0.00044072

0.0009855

0.00174112

0.0027035

0.00386856

0.00523221

166

0.00010821

0.00043023

0.00096212

0.00169993

0.00263975

0.00377763

0.00510964

168

0.00010566

0.00042011

0.00093955

0.00166019

0.00257823

0.00368987

0.00499132

170

0.0001032

0.00041034

0.00091777

0.00162182

0.00251883

0.00360513

0.00487706

172

0.00010082

0.00040091

0.00089674

0.00158477

0.00246146

0.00352327

0.00476668

174

9.8519E-05

0.0003918

0.00087642

0.00154897

0.00240603

0.00344418

0.00466

176

9.6299E-05

0.00038299

0.00085679

0.00151437

0.00235244

0.00336771

0.00455686

178

9.4153E-05

0.00037448

0.0008378

0.00148092

0.00230063

0.00329376

0.00445711

180

9.2078E-05

0.00036626

0.00081944

0.00144856

0.00225051

0.00322222

0.0043606

182

9.0071E-05

0.00035829

0.00080168

0.00141725

0.00220202

0.00315299

0.00426718

184

8.8129E-05

0.00035059

0.00078449

0.00138695

0.00215507

0.00308596

0.00417674

186

8.6249E-05

0.00034313

0.00076785

0.00135761

0.0021096

0.00302105

0.00408914

188

8.4429E-05

0.00033591

0.00075173

0.00132919

0.00206556

0.00295816

0.00400426

190

8.2665E-05

0.00032891

0.00073611

0.00130165

0.00202289

0.00289722

0.003922

192

8.0956E-05

0.00032213

0.00072098

0.00127496

0.00198152

0.00283813

0.00384225

194

7.93E-05

0.00031556

0.0007063

0.00124908

0.00194141

0.00278084

0.0037649

196

7.7694E-05

0.00030918

0.00069207

0.00122398

0.00190251

0.00272527

0.00368987

198

7.6136E-05

0.000303

0.00067827

0.00119963

0.00186476

0.00267134

0.00361706

200

7.4625E-05

0.000297

0.00066488

0.001176

0.00182813

0.002619

0.00354638

 

           

            Example of using the table:

            Find the probability of a plum occurring exactly twice on a payline of a 3-reel slot machine with 64 stops on each reel, having 7 plums on each reel.

            We look at the intersection of row t = 64 with column c = 7 and find the probability P = 0.03196335= 3.196335%.