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%.