3-reel slot machines

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

         1.1  Event   A specific symbol three times

            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 three times on a payline 

c

1

2

3

4

5

6

7

t

20

0.000125

0.001

0.003375

0.008

0.015625

0.027

0.042875

22

9.3914E-05

0.00075131

0.00253569

0.00601052

0.01173929

0.0202855

0.03221262

24

7.2338E-05

0.0005787

0.00195313

0.00462963

0.00904225

0.015625

0.02481192

26

5.6896E-05

0.00045517

0.00153619

0.00364133

0.00711197

0.01228949

0.01951525

28

4.5554E-05

0.00036443

0.00122996

0.00291545

0.00569424

0.00983965

0.015625

30

3.7037E-05

0.0002963

0.001

0.00237037

0.00462963

0.008

0.0127037

32

3.0518E-05

0.00024414

0.00082397

0.00195313

0.0038147

0.0065918

0.01046753

34

2.5443E-05

0.00020354

0.00068695

0.00162833

0.00318034

0.00549562

0.00872685

36

2.1433E-05

0.00017147

0.0005787

0.00137174

0.00267918

0.00462963

0.00735168

38

1.8224E-05

0.00014579

0.00049205

0.00116635

0.00227803

0.00393643

0.00625091

40

1.5625E-05

0.000125

0.00042188

0.001

0.00195313

0.003375

0.00535938

42

1.3497E-05

0.00010798

0.00036443

0.00086384

0.00168718

0.00291545

0.00462963

44

1.1739E-05

9.3914E-05

0.00031696

0.00075131

0.00146741

0.00253569

0.00402658

46

1.0274E-05

8.219E-05

0.00027739

0.00065752

0.00128421

0.00221912

0.00352388

48

9.0422E-06

7.2338E-05

0.00024414

0.0005787

0.00113028

0.00195313

0.00310149

50

0.000008

0.000064

0.000216

0.000512

0.001

0.001728

0.002744

52

7.112E-06

5.6896E-05

0.00019202

0.00045517

0.000889

0.00153619

0.00243941

54

6.3507E-06

5.0805E-05

0.00017147

0.00040644

0.00079383

0.00137174

0.00217828

56

5.6942E-06

4.5554E-05

0.00015374

0.00036443

0.00071178

0.00122996

0.00195313

58

5.1253E-06

4.1002E-05

0.00013838

0.00032802

0.00064066

0.00110706

0.00175796

60

4.6296E-06

3.7037E-05

0.000125

0.0002963

0.0005787

0.001

0.00158796

62

4.1959E-06

3.3567E-05

0.00011329

0.00026854

0.00052449

0.00090631

0.00143919

64

3.8147E-06

3.0518E-05

0.000103

0.00024414

0.00047684

0.00082397

0.00130844

66

3.4783E-06

2.7826E-05

9.3914E-05

0.00022261

0.00043479

0.00075131

0.00119306

68

3.1803E-06

2.5443E-05

8.5869E-05

0.00020354

0.00039754

0.00068695

0.00109086

70

2.9155E-06

2.3324E-05

7.8717E-05

0.00018659

0.00036443

0.00062974

0.001

72

2.6792E-06

2.1433E-05

7.2338E-05

0.00017147

0.0003349

0.0005787

0.00091896

74

2.4678E-06

1.9742E-05

6.663E-05

0.00015794

0.00030847

0.00053304

0.00084645

76

2.278E-06

1.8224E-05

6.1507E-05

0.00014579

0.00028475

0.00049205

0.00078136

78

2.1073E-06

1.6858E-05

5.6896E-05

0.00013486

0.00026341

0.00045517

0.00072279

80

1.9531E-06

1.5625E-05

5.2734E-05

0.000125

0.00024414

0.00042188

0.00066992

82

1.8137E-06

1.4509E-05

4.8969E-05

0.00011607

0.00022671

0.00039175

0.00062209

84

1.6872E-06

1.3497E-05

4.5554E-05

0.00010798

0.0002109

0.00036443

0.0005787

86

1.5722E-06

1.2578E-05

4.2449E-05

0.00010062

0.00019652

0.00033959

0.00053926

88

1.4674E-06

1.1739E-05

3.962E-05

9.3914E-05

0.00018343

0.00031696

0.00050332

90

1.3717E-06

1.0974E-05

3.7037E-05

8.7791E-05

0.00017147

0.0002963

0.00047051

92

1.2842E-06

1.0274E-05

3.4674E-05

8.219E-05

0.00016053

0.00027739

0.00044048

94

1.204E-06

9.6318E-06

3.2507E-05

7.7054E-05

0.0001505

0.00026006

0.00041296

96

1.1303E-06

9.0422E-06

3.0518E-05

7.2338E-05

0.00014129

0.00024414

0.00038769

98

1.0625E-06

8.4999E-06

2.8687E-05

6.7999E-05

0.00013281

0.0002295

0.00036443

100

0.000001

0.000008

0.000027

0.000064

0.000125

0.000216

0.000343

102

9.4232E-07

7.5386E-06

2.5443E-05

6.0309E-05

0.00011779

0.00020354

0.00032322

104

8.89E-07

7.112E-06

2.4003E-05

5.6896E-05

0.00011112

0.00019202

0.00030493

106

8.3962E-07

6.717E-06

2.267E-05

5.3736E-05

0.00010495

0.00018136

0.00028799

108

7.9383E-07

6.3507E-06

2.1433E-05

5.0805E-05

9.9229E-05

0.00017147

0.00027228

110

7.5131E-07

6.0105E-06

2.0285E-05

4.8084E-05

9.3914E-05

0.00016228

0.0002577

112

7.1178E-07

5.6942E-06

1.9218E-05

4.5554E-05

8.8973E-05

0.00015374

0.00024414

114

6.7497E-07

5.3998E-06

1.8224E-05

4.3198E-05

8.4371E-05

0.00014579

0.00023152

116

6.4066E-07

5.1253E-06

1.7298E-05

4.1002E-05

8.0082E-05

0.00013838

0.00021975

118

6.0863E-07

4.869E-06

1.6433E-05

3.8952E-05

7.6079E-05

0.00013146

0.00020876

120

5.787E-07

4.6296E-06

1.5625E-05

3.7037E-05

7.2338E-05

0.000125

0.0001985

122

5.5071E-07

4.4057E-06

1.4869E-05

3.5245E-05

6.8838E-05

0.00011895

0.00018889

124

5.2449E-07

4.1959E-06

1.4161E-05

3.3567E-05

6.5561E-05

0.00011329

0.0001799

126

4.9991E-07

3.9992E-06

1.3497E-05

3.1994E-05

6.2488E-05

0.00010798

0.00017147

128

4.7684E-07

3.8147E-06

1.2875E-05

3.0518E-05

5.9605E-05

0.000103

0.00016356

130

4.5517E-07

3.6413E-06

1.2289E-05

2.9131E-05

5.6896E-05

9.8316E-05

0.00015612

132

4.3479E-07

3.4783E-06

1.1739E-05

2.7826E-05

5.4349E-05

9.3914E-05

0.00014913

134

4.1561E-07

3.3249E-06

1.1221E-05

2.6599E-05

5.1951E-05

8.9772E-05

0.00014255

136

3.9754E-07

3.1803E-06

1.0734E-05

2.5443E-05

4.9693E-05

8.5869E-05

0.00013636

138

3.8051E-07

3.0441E-06

1.0274E-05

2.4352E-05

4.7563E-05

8.219E-05

0.00013051

140

3.6443E-07

2.9155E-06

9.8397E-06

2.3324E-05

4.5554E-05

7.8717E-05

0.000125

142

3.4925E-07

2.794E-06

9.4297E-06

2.2352E-05

4.3656E-05

7.5438E-05

0.00011979

144

3.349E-07

2.6792E-06

9.0422E-06

2.1433E-05

4.1862E-05

7.2338E-05

0.00011487

148

3.0847E-07

2.4678E-06

8.3287E-06

1.9742E-05

3.8559E-05

6.663E-05

0.00010581

150

2.963E-07

2.3704E-06

0.000008

1.8963E-05

3.7037E-05

0.000064

0.00010163

152

2.8475E-07

2.278E-06

7.6883E-06

1.8224E-05

3.5594E-05

6.1507E-05

9.767E-05

154

2.738E-07

2.1904E-06

7.3927E-06

1.7523E-05

3.4225E-05

5.9141E-05

9.3914E-05

156

2.6341E-07

2.1073E-06

7.112E-06

1.6858E-05

3.2926E-05

5.6896E-05

9.0348E-05

158

2.5353E-07

2.0282E-06

6.8453E-06

1.6226E-05

3.1691E-05

5.4762E-05

8.6961E-05

160

2.4414E-07

1.9531E-06

6.5918E-06

1.5625E-05

3.0518E-05

5.2734E-05

8.374E-05

162

2.3521E-07

1.8817E-06

6.3507E-06

1.5053E-05

2.9401E-05

5.0805E-05

8.0677E-05

164

2.2671E-07

1.8137E-06

6.1211E-06

1.4509E-05

2.8339E-05

4.8969E-05

7.7761E-05

166

2.1861E-07

1.7489E-06

5.9025E-06

1.3991E-05

2.7327E-05

4.722E-05

7.4984E-05

168

2.109E-07

1.6872E-06

5.6942E-06

1.3497E-05

2.6362E-05

4.5554E-05

7.2338E-05

170

2.0354E-07

1.6283E-06

5.4956E-06

1.3027E-05

2.5443E-05

4.3965E-05

6.9815E-05

172

1.9652E-07

1.5722E-06

5.3061E-06

1.2578E-05

2.4565E-05

4.2449E-05

6.7408E-05

174

1.8982E-07

1.5186E-06

5.1253E-06

1.2149E-05

2.3728E-05

4.1002E-05

6.511E-05

176

1.8343E-07

1.4674E-06

4.9525E-06

1.1739E-05

2.2928E-05

3.962E-05

6.2915E-05

178

1.7731E-07

1.4185E-06

4.7874E-06

1.1348E-05

2.2164E-05

3.83E-05

6.0818E-05

180

1.7147E-07

1.3717E-06

4.6296E-06

1.0974E-05

2.1433E-05

3.7037E-05

5.8813E-05

182

1.6588E-07

1.327E-06

4.4787E-06

1.0616E-05

2.0735E-05

3.5829E-05

5.6896E-05

184

1.6053E-07

1.2842E-06

4.3342E-06

1.0274E-05

2.0066E-05

3.4674E-05

5.5061E-05

186

1.554E-07

1.2432E-06

4.1959E-06

9.9458E-06

1.9425E-05

3.3567E-05

5.3303E-05

188

1.505E-07

1.204E-06

4.0634E-06

9.6318E-06

1.8812E-05

3.2507E-05

5.162E-05

190

1.4579E-07

1.1664E-06

3.9364E-06

9.3308E-06

1.8224E-05

3.1491E-05

5.0007E-05

192

1.4129E-07

1.1303E-06

3.8147E-06

9.0422E-06

1.7661E-05

3.0518E-05

4.8461E-05

194

1.3696E-07

1.0957E-06

3.6979E-06

8.7655E-06

1.712E-05

2.9583E-05

4.6977E-05

196

1.3281E-07

1.0625E-06

3.5859E-06

8.4999E-06

1.6601E-05

2.8687E-05

4.5554E-05

198

1.2883E-07

1.0306E-06

3.4783E-06

8.2449E-06

1.6103E-05

2.7826E-05

4.4187E-05

200

1.25E-07

0.000001

3.375E-06

0.000008

1.5625E-05

0.000027

4.2875E-05

             Note: The notation of a number using letter E called scientific notation applies for very low numbers with a large number of decimal places. To convert a number from scientific notation to its standard decimal notation, we must move the decimal point to the left the number of decimal places indicated by the number written after E-. For instance, 1.25E-07 converts to 0.000000125, which means a 0.0000125% probability.

                   Examples of using the table:           

           1)  Find the probability of a cherry occurring three times on a payline of a 3-reel slot machine with 82 stops on each reel, having 4 cherries on each reel.

            We look at the intersection of row t = 82 with column c = 4 and find the probability P = 0.00011607 = 0.011607%.

            2)  Find the probability of a red seven occurring three times on a payline of a 3-reel slot machine with 51 stops on each reel, having 5 red sevens on each reel.

            We observe that 50 < 51 < 52.

            We look at the intersection of row t = 50 with column c = 5 and find the first probability as 0.001.  We look at the intersection of row t = 50 with column c = 5 and find the second probability as 0.000889.

            The sought probability is between 0.000889 and 0.001. We could choose for instance P = 0.00095 = 0.095% as an approximated probability.