3-reel slot machines

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

       1.8  A specific symbol at least twice

            The probability of this events 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.

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

c

1

2

3

4

5

6

7

t

20

0.00725

0.028

0.06075

0.104

0.15625

0.216

0.28175

22

0.00601052

0.02329076

0.05071375

0.08715252

0.13148009

0.1825695

0.23929376

24

0.00506366

0.01967593

0.04296875

0.07407407

0.11212384

0.15625

0.20558449

26

0.00432408

0.01684115

0.03686846

0.06372326

0.0967228

0.13518434

0.17842513

28

0.00373542

0.01457726

0.03197886

0.05539359

0.08427478

0.1180758

0.15625

30

0.00325926

0.01274074

0.028

0.04859259

0.07407407

0.104

0.13792593

32

0.00286865

0.01123047

0.02471924

0.04296875

0.06561279

0.09228516

0.12261963

34

0.00254427

0.00997354

0.0219825

0.03826583

0.05851822

0.08243436

0.10970894

36

0.00227195

0.00891632

0.01967593

0.03429355

0.052512

0.07407407

0.09872257

38

0.00204111

0.00801866

0.01771395

0.0309083

0.047383

0.06691938

0.08929873

40

0.00184375

0.00725

0.01603125

0.028

0.04296875

0.06075

0.08115625

42

0.00167369

0.00658676

0.01457726

0.02548321

0.03914264

0.05539359

0.07407407

44

0.00152611

0.00601052

0.01331236

0.02329076

0.03580485

0.05071375

0.0678766

46

0.00139722

0.0055067

0.01220515

0.02136928

0.03287581

0.04660146

0.06242295

48

0.001284

0.00506366

0.01123047

0.01967593

0.03029152

0.04296875

0.0575991

50

0.001184

0.004672

0.010368

0.018176

0.028

0.039744

0.053312

52

0.00109524

0.00432408

0.00960116

0.01684115

0.02595869

0.03686846

0.04948509

54

0.00101611

0.00401362

0.00891632

0.01564802

0.0241325

0.03429355

0.04605497

56

0.00094524

0.00373542

0.0083022

0.01457726

0.02249226

0.03197886

0.04296875

58

0.00088154

0.00348518

0.0077494

0.01361269

0.02101357

0.02989052

0.04018205

60

0.00082407

0.00325926

0.00725

0.01274074

0.01967593

0.028

0.03765741

62

0.00077205

0.00305461

0.00679735

0.01194992

0.01846195

0.02628311

0.03536303

64

0.00072479

0.00286865

0.0063858

0.01123047

0.01735687

0.02471924

0.03327179

66

0.00068175

0.00269917

0.00601052

0.01057406

0.01634805

0.02329076

0.03136044

68

0.00064243

0.00254427

0.00566736

0.00997354

0.01542464

0.0219825

0.02960895

70

0.00060641

0.00240233

0.00535277

0.00942274

0.01457726

0.02078134

0.028

72

0.00057335

0.00227195

0.00506366

0.00891632

0.0137978

0.01967593

0.02651856

74

0.00054291

0.0021519

0.00479735

0.00844965

0.01307919

0.01865635

0.02515152

76

0.00051483

0.00204111

0.0045515

0.00801866

0.01241526

0.01771395

0.02388741

78

0.00048888

0.00193867

0.00432408

0.00761982

0.0118006

0.01684115

0.02271616

80

0.00046484

0.00184375

0.00411328

0.00725

0.01123047

0.01603125

0.02162891

82

0.00044254

0.00175563

0.00391753

0.00690646

0.01070066

0.01527836

0.02061781

84

0.0004218

0.00167369

0.00373542

0.00658676

0.01020746

0.01457726

0.01967593

86

0.00040248

0.00159734

0.00356572

0.00628875

0.00974757

0.0139233

0.01879709

88

0.00038446

0.00152611

0.00340733

0.00601052

0.00931806

0.01331236

0.01797579

90

0.00036763

0.00145953

0.00325926

0.00575034

0.00891632

0.01274074

0.01720713

92

0.00035187

0.00139722

0.00312063

0.0055067

0.00854001

0.01220515

0.01648671

94

0.00033711

0.00133882

0.00299067

0.00527821

0.00818701

0.01170261

0.01581056

96

0.00032326

0.001284

0.00286865

0.00506366

0.00785545

0.01123047

0.01517515

98

0.00031024

0.00123248

0.00275395

0.00486192

0.00754363

0.01078632

0.01457726

100

0.000298

0.001184

0.002646

0.004672

0.00725

0.010368

0.014014

102

0.00028647

0.00113833

0.00254427

0.00449299

0.00697319

0.00997354

0.01348275

104

0.00027559

0.00109524

0.0024483

0.00432408

0.00671192

0.00960116

0.01298112

106

0.00026532

0.00105456

0.00235765

0.00416451

0.00646507

0.00924925

0.01250697

108

0.00025561

0.00101611

0.00227195

0.00401362

0.00623158

0.00891632

0.01205831

110

0.00024643

0.00097971

0.00219083

0.00387077

0.00601052

0.00860105

0.01163336

112

0.00023773

0.00094524

0.00211399

0.00373542

0.00580101

0.0083022

0.01123047

114

0.00022949

0.00091256

0.00204111

0.00360705

0.00560226

0.00801866

0.01084814

116

0.00022167

0.00088154

0.00197194

0.00348518

0.00541356

0.0077494

0.010485

118

0.00021424

0.00085208

0.00190623

0.00336938

0.00523423

0.00749346

0.01013979

120

0.00020718

0.00082407

0.00184375

0.00325926

0.00506366

0.00725

0.00981134

122

0.00020046

0.00079742

0.00178429

0.00315445

0.00490129

0.00701821

0.00949859

124

0.00019406

0.00077205

0.00172766

0.00305461

0.00474661

0.00679735

0.00920056

126

0.00018796

0.00074786

0.00167369

0.00295944

0.00459914

0.00658676

0.00891632

128

0.00018215

0.00072479

0.0016222

0.00286865

0.00445843

0.0063858

0.00864506

130

0.0001766

0.00070278

0.00157305

0.00278198

0.00432408

0.0061939

0.00838598

132

0.00017131

0.00068175

0.00152611

0.00269917

0.00419571

0.00601052

0.00813837

134

0.00016624

0.00066165

0.00148123

0.00262

0.00407297

0.00583516

0.00790157

136

0.0001614

0.00064243

0.00143831

0.00254427

0.00395555

0.00566736

0.00767495

138

0.00015677

0.00062403

0.00139722

0.00247177

0.00384312

0.0055067

0.00745794

140

0.00015233

0.00060641

0.00135787

0.00240233

0.00373542

0.00535277

0.00725

142

0.00014808

0.00058953

0.00132016

0.00233578

0.00363219

0.0052052

0.00705064

144

0.00014401

0.00057335

0.001284

0.00227195

0.00353317

0.00506366

0.00685938

148

0.00013634

0.00054291

0.00121599

0.0021519

0.00334691

0.00479735

0.00649949

150

0.00013274

0.00052859

0.001184

0.00209541

0.00325926

0.004672

0.00633007

152

0.00012928

0.00051483

0.00115325

0.00204111

0.003175

0.0045515

0.00616719

154

0.00012595

0.00050161

0.00112369

0.0019889

0.00309397

0.0044356

0.00601052

156

0.00012275

0.00048888

0.00109524

0.00193867

0.003016

0.00432408

0.00585974

158

0.00011967

0.00047664

0.00106787

0.00189032

0.00294094

0.0042167

0.00571456

160

0.0001167

0.00046484

0.0010415

0.00184375

0.00286865

0.00411328

0.00557471

162

0.00011384

0.00045348

0.00101611

0.00179888

0.00279899

0.00401362

0.00543993

164

0.00011109

0.00044254

0.00099162

0.00175563

0.00273184

0.00391753

0.00530997

166

0.00010843

0.00043198

0.00096802

0.00171392

0.00266708

0.00382485

0.00518462

168

0.00010587

0.0004218

0.00094524

0.00167369

0.00260459

0.00373542

0.00506366

170

0.0001034

0.00041197

0.00092326

0.00163485

0.00254427

0.00364909

0.00494688

172

0.00010101

0.00040248

0.00090204

0.00159734

0.00248602

0.00356572

0.00483409

174

9.8709E-05

0.00039332

0.00088154

0.00156112

0.00242975

0.00348518

0.00472511

176

9.6482E-05

0.00038446

0.00086174

0.00152611

0.00237537

0.00340733

0.00461978

178

9.433E-05

0.0003759

0.00084259

0.00149226

0.0023228

0.00333206

0.00451793

180

9.225E-05

0.00036763

0.00082407

0.00145953

0.00227195

0.00325926

0.00441941

182

9.0237E-05

0.00035962

0.00080616

0.00142787

0.00222275

0.00318882

0.00432408

184

8.829E-05

0.00035187

0.00078883

0.00139722

0.00217513

0.00312063

0.0042318

186

8.6404E-05

0.00034437

0.00077205

0.00136755

0.00212903

0.00305461

0.00414244

188

8.4579E-05

0.00033711

0.00075579

0.00133882

0.00208438

0.00299067

0.00405588

190

8.2811E-05

0.00033008

0.00074005

0.00131098

0.00204111

0.00292871

0.00397201

192

8.1098E-05

0.00032326

0.00072479

0.001284

0.00199918

0.00286865

0.00389071

194

7.9437E-05

0.00031665

0.00071

0.00125784

0.00195853

0.00281043

0.00381188

196

7.7827E-05

0.00031024

0.00069566

0.00123248

0.00191911

0.00275395

0.00373542

198

7.6265E-05

0.00030403

0.00068175

0.00120788

0.00188086

0.00269917

0.00366124

200

0.00007475

0.000298

0.00066825

0.001184

0.00184375

0.002646

0.00358925

 

            Example of using the table:

            Find the probability of a cherry occurring at least twice on a payline of a 3-reel slot machine with 34 stops on each reel, having 3 cherries on each reel.

            We look in the table, at the intersection of row t = 34 with column c = 3 and find the probability P = 0.0219825 = 2.19825%.