16-reel slot machines

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

1.7  A specific symbol at least three times

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 three times on a payline

 c 1 2 3 4 5 6 7 t 20 0.00048125 0.0037 0.01198125 0.0272 0.05078125 0.0837 0.12648125 22 0.000362851 0.002800355 0.009105423 0.020763609 0.038953111 0.064544772 0.098102076 24 0.00028031 0.002170139 0.007080078 0.016203704 0.030517578 0.05078125 0.077537254 26 0.000221018 0.001715626 0.005612986 0.012884703 0.024344823 0.040649837 0.062298677 28 0.000177335 0.001379633 0.004524482 0.010412328 0.019726481 0.033033111 0.05078125 30 0.000144444 0.001125926 0.0037 0.008533333 0.016203704 0.0272 0.041922222 32 0.000119209 0.000930786 0.003064156 0.007080078 0.01347065 0.022659302 0.035000801 34 9.95259E-05 0.000778247 0.002565971 0.005938626 0.011318261 0.019073047 0.029517277 36 8.39478E-05 0.000657293 0.002170139 0.005029721 0.009600409 0.016203704 0.025118241 38 7.14582E-05 0.000560155 0.001851678 0.004297082 0.008212894 0.013881109 0.021549194 40 6.13281E-05 0.00048125 0.001592578 0.0037 0.007080078 0.01198125 0.018623828 42 5.30257E-05 0.000416493 0.001379633 0.00320854 0.006146166 0.010412328 0.016203704 44 4.61568E-05 0.000362851 0.001203011 0.002800355 0.005369393 0.009105423 0.014184535 46 4.04247E-05 0.000318038 0.001055287 0.002458539 0.004718081 0.008008119 0.012486778 48 3.56038E-05 0.00028031 0.000930786 0.002170139 0.00416791 0.007080078 0.011049059 50 0.00003152 0.00024832 0.00082512 0.00192512 0.0037 0.00628992 0.00982352 52 2.80376E-05 0.000221018 0.000734858 0.001715626 0.003299544 0.005612986 0.008772479 54 2.50498E-05 0.000197576 0.000657293 0.001535448 0.00295482 0.005029721 0.007865995 56 2.24719E-05 0.000177335 0.000590269 0.001379633 0.002656466 0.004524482 0.007080078 58 2.02359E-05 0.000159767 0.000532055 0.001244201 0.002396943 0.004084657 0.006395354 60 1.8287E-05 0.000144444 0.00048125 0.001125926 0.002170139 0.0037 0.005796065 62 1.65806E-05 0.00013102 0.000436712 0.001022175 0.001971057 0.003362133 0.005269304 64 1.508E-05 0.000119209 0.000397503 0.000930786 0.00179559 0.003064156 0.004804432 66 1.37551E-05 0.000108776 0.000362851 0.000849972 0.001640339 0.002800355 0.00439263 68 1.2581E-05 9.95259E-05 0.000332111 0.000778247 0.001502476 0.002565971 0.004026542 70 1.15369E-05 9.12953E-05 0.000304748 0.000714369 0.001379633 0.002357018 0.0037 72 1.06051E-05 8.39478E-05 0.00028031 0.000657293 0.001269821 0.002170139 0.00340781 74 9.77104E-06 7.7368E-05 0.000258416 0.000606138 0.001171357 0.002002496 0.003145574 76 9.02219E-06 7.14582E-05 0.000238743 0.000560155 0.001082813 0.001851678 0.002909552 78 8.34795E-06 6.61353E-05 0.000221018 0.000518708 0.00100297 0.001715626 0.002696551 80 7.73926E-06 6.13281E-05 0.000205005 0.00048125 0.000930786 0.001592578 0.002503833 82 7.18833E-06 5.69758E-05 0.000190502 0.000447313 0.000865364 0.001481017 0.002329041 84 6.68847E-06 5.30257E-05 0.000177335 0.000416493 0.000805931 0.001379633 0.002170139 86 6.23391E-06 4.94325E-05 0.000165354 0.00038844 0.000751817 0.001287293 0.002025363 88 5.81962E-06 4.61568E-05 0.000154428 0.000362851 0.00070244 0.001203011 0.001893178 90 5.44124E-06 4.31642E-05 0.000144444 0.00033946 0.000657293 0.001125926 0.001772245 92 5.09497E-06 4.04247E-05 0.000135303 0.000318038 0.000615933 0.001055287 0.001661393 94 4.77746E-06 3.79123E-05 0.000126917 0.00029838 0.000577971 0.000990434 0.001559592 96 4.4858E-06 3.56038E-05 0.000119209 0.00028031 0.000543065 0.000930786 0.001465939 98 4.2174E-06 3.3479E-05 0.000112114 0.000263669 0.000510913 0.000875832 0.001379633 100 0.00000397 0.00003152 0.00010557 0.00024832 0.00048125 0.00082512 0.00129997 102 3.74157E-06 2.97109E-05 9.95259E-05 0.000234139 0.000453839 0.000778247 0.001226322 104 3.53034E-06 2.80376E-05 9.39344E-05 0.000221018 0.000428471 0.000734858 0.001158131 106 3.33471E-06 2.64876E-05 8.87541E-05 0.000208859 0.000404958 0.000694634 0.001094903 108 3.15328E-06 2.50498E-05 8.39478E-05 0.000197576 0.000383134 0.000657293 0.001036194 110 2.98477E-06 2.37142E-05 7.94823E-05 0.000187091 0.000362851 0.00062258 0.000981606 112 2.82806E-06 2.24719E-05 7.5328E-05 0.000177335 0.000343974 0.000590269 0.000930786 114 2.68212E-06 2.13149E-05 7.14582E-05 0.000168246 0.000326384 0.000560155 0.000883413 116 2.54606E-06 2.02359E-05 6.7849E-05 0.000159767 0.000309973 0.000532055 0.000839201 118 2.41905E-06 1.92286E-05 6.44788E-05 0.000151848 0.000294644 0.000505803 0.000797889 120 2.30035E-06 1.8287E-05 6.13281E-05 0.000144444 0.00028031 0.00048125 0.000759245 122 2.18929E-06 1.74059E-05 5.83794E-05 0.000137514 0.00026689 0.00045826 0.000723056 124 2.08526E-06 1.65806E-05 5.56168E-05 0.00013102 0.000254313 0.000436712 0.00068913 126 1.98772E-06 1.58066E-05 5.30257E-05 0.000124929 0.000242514 0.000416493 0.000657293 128 1.89617E-06 1.508E-05 5.05932E-05 0.000119209 0.000231434 0.000397503 0.000627387 130 1.81016E-06 1.43973E-05 4.83071E-05 0.000113834 0.000221018 0.000379651 0.000599268 132 1.72927E-06 1.37551E-05 4.61568E-05 0.000108776 0.000211218 0.000362851 0.000572804 134 1.65313E-06 1.31506E-05 4.41322E-05 0.000104014 0.000201989 0.000347028 0.000547876 136 1.5814E-06 1.2581E-05 4.22242E-05 9.95259E-05 0.00019329 0.000332111 0.000524373 138 1.51376E-06 1.20439E-05 4.04247E-05 9.52922E-05 0.000185084 0.000318038 0.000502195 140 1.44992E-06 1.15369E-05 3.87261E-05 9.12953E-05 0.000177335 0.000304748 0.00048125 142 1.38962E-06 1.10579E-05 3.71212E-05 8.75188E-05 0.000170013 0.000292188 0.000461454 144 1.33261E-06 1.06051E-05 3.56038E-05 8.39478E-05 0.000163088 0.00028031 0.000442728 148 1.22763E-06 9.77104E-06 3.28084E-05 7.7368E-05 0.000150328 0.000258416 0.00040821 150 1.17926E-06 9.38667E-06 0.00003152 7.43348E-05 0.000144444 0.00024832 0.00039229 152 1.13339E-06 9.02219E-06 3.02982E-05 7.14582E-05 0.000138864 0.000238743 0.000377188 154 1.08988E-06 8.67635E-06 2.91387E-05 6.87281E-05 0.000133568 0.000229653 0.000362851 156 1.04856E-06 8.34795E-06 2.80376E-05 6.61353E-05 0.000128537 0.000221018 0.000349231 158 1.0093E-06 8.03593E-06 2.69913E-05 6.36712E-05 0.000123756 0.000212811 0.000336285 160 9.71985E-07 7.73926E-06 2.59964E-05 6.13281E-05 0.000119209 0.000205005 0.00032397 162 9.36482E-07 7.45701E-06 2.50498E-05 5.90986E-05 0.000114882 0.000197576 0.000312249 164 9.02688E-07 7.18833E-06 2.41486E-05 5.69758E-05 0.000110762 0.000190502 0.000301087 166 8.70501E-07 6.9324E-06 2.32902E-05 5.49535E-05 0.000106837 0.000183761 0.000290451 168 8.39825E-07 6.68847E-06 2.24719E-05 5.30257E-05 0.000103095 0.000177335 0.00028031 170 8.10575E-07 6.45586E-06 2.16916E-05 5.11871E-05 9.95259E-05 0.000171205 0.000270635 172 7.82667E-07 6.23391E-06 2.09469E-05 4.94325E-05 9.61195E-05 0.000165354 0.0002614 174 7.56025E-07 6.02202E-06 2.02359E-05 4.77572E-05 9.28667E-05 0.000159767 0.000252581 176 7.30579E-07 5.81962E-06 1.95568E-05 4.61568E-05 8.97591E-05 0.000154428 0.000244154 178 7.06263E-07 5.62619E-06 1.89077E-05 4.4627E-05 8.67886E-05 0.000149325 0.000236098 180 6.83013E-07 5.44124E-06 1.8287E-05 4.31642E-05 8.39478E-05 0.000144444 0.000228392 182 6.60773E-07 5.26431E-06 1.76932E-05 4.17645E-05 8.12295E-05 0.000139774 0.000221018 184 6.39488E-07 5.09497E-06 1.71249E-05 4.04247E-05 7.86274E-05 0.000135303 0.000213958 186 6.19108E-07 4.93281E-06 1.65806E-05 3.91417E-05 7.61352E-05 0.00013102 0.000207196 188 5.99585E-07 4.77746E-06 1.60591E-05 3.79123E-05 7.37473E-05 0.000126917 0.000200715 190 5.80873E-07 4.62857E-06 1.55593E-05 3.67339E-05 7.14582E-05 0.000122982 0.000194502 192 5.62933E-07 4.4858E-06 1.508E-05 3.56038E-05 6.92628E-05 0.000119209 0.000188543 194 5.45723E-07 4.34884E-06 1.46202E-05 3.45197E-05 6.71565E-05 0.000115589 0.000182824 196 5.29208E-07 4.2174E-06 1.41789E-05 3.3479E-05 6.51346E-05 0.000112114 0.000177335 198 5.13353E-07 4.09121E-06 1.37551E-05 3.24798E-05 6.31932E-05 0.000108776 0.000172063 200 4.98125E-07 0.00000397 1.33481E-05 0.00003152 6.13281E-05 0.00010557 0.000166998