3-reel slot machines

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

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%.