3-reel slot machines

2. Case B – different numbers of stops and symbol distributions on the reels

2.8  A specific symbol at least twice

The probability of this event is , where  are the basic probabilities of that symbol occurring on the reels respectively. In the particular case when two of the three reels have the same number of stops and the same distribution of symbols, the probability of the measured event is , where  is the basic probability of that symbol occurring on one of the two similar reels and  is the basic probability of that symbol occurring on the third reel. This particular formula returns the following table of values, where the values of the basic probabilities are listed in increments of 0.005, ranging from 0.005 to 0.100, and each table lists seven values of .

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

Table 1:   from 0.005 to 0.035

 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.005 0.00007475 0.0001245 0.00017425 0.000224 0.00027375 0.0003235 0.00037325 0.01 0.000199 0.000298 0.000397 0.000496 0.000595 0.000694 0.000793 0.015 0.00037275 0.0005205 0.00066825 0.000816 0.00096375 0.0011115 0.00125925 0.02 0.000596 0.000792 0.000988 0.001184 0.00138 0.001576 0.001772 0.025 0.00086875 0.0011125 0.00135625 0.0016 0.00184375 0.0020875 0.00233125 0.03 0.001191 0.001482 0.001773 0.002064 0.002355 0.002646 0.002937 0.035 0.00156275 0.0019005 0.00223825 0.002576 0.00291375 0.0032515 0.00358925 0.04 0.001984 0.002368 0.002752 0.003136 0.00352 0.003904 0.004288 0.045 0.00245475 0.0028845 0.00331425 0.003744 0.00417375 0.0046035 0.00503325 0.05 0.002975 0.00345 0.003925 0.0044 0.004875 0.00535 0.005825 0.055 0.00354475 0.0040645 0.00458425 0.005104 0.00562375 0.0061435 0.00666325 0.06 0.004164 0.004728 0.005292 0.005856 0.00642 0.006984 0.007548 0.065 0.00483275 0.0054405 0.00604825 0.006656 0.00726375 0.0078715 0.00847925 0.07 0.005551 0.006202 0.006853 0.007504 0.008155 0.008806 0.009457 0.075 0.00631875 0.0070125 0.00770625 0.0084 0.00909375 0.0097875 0.01048125 0.08 0.007136 0.007872 0.008608 0.009344 0.01008 0.010816 0.011552 0.085 0.00800275 0.0087805 0.00955825 0.010336 0.01111375 0.0118915 0.01266925 0.09 0.008919 0.009738 0.010557 0.011376 0.012195 0.013014 0.013833 0.095 0.00988475 0.0107445 0.01160425 0.012464 0.01332375 0.0141835 0.01504325 0.1 0.0109 0.0118 0.0127 0.0136 0.0145 0.0154 0.0163

Table 2:   from 0.040 to 0.070

 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.005 0.000423 0.00047275 0.0005225 0.00057225 0.000622 0.00067175 0.0007215 0.01 0.000892 0.000991 0.00109 0.001189 0.001288 0.001387 0.001486 0.015 0.001407 0.00155475 0.0017025 0.00185025 0.001998 0.00214575 0.0022935 0.02 0.001968 0.002164 0.00236 0.002556 0.002752 0.002948 0.003144 0.025 0.002575 0.00281875 0.0030625 0.00330625 0.00355 0.00379375 0.0040375 0.03 0.003228 0.003519 0.00381 0.004101 0.004392 0.004683 0.004974 0.035 0.003927 0.00426475 0.0046025 0.00494025 0.005278 0.00561575 0.0059535 0.04 0.004672 0.005056 0.00544 0.005824 0.006208 0.006592 0.006976 0.045 0.005463 0.00589275 0.0063225 0.00675225 0.007182 0.00761175 0.0080415 0.05 0.0063 0.006775 0.00725 0.007725 0.0082 0.008675 0.00915 0.055 0.007183 0.00770275 0.0082225 0.00874225 0.009262 0.00978175 0.0103015 0.06 0.008112 0.008676 0.00924 0.009804 0.010368 0.010932 0.011496 0.065 0.009087 0.00969475 0.0103025 0.01091025 0.011518 0.01212575 0.0127335 0.07 0.010108 0.010759 0.01141 0.012061 0.012712 0.013363 0.014014 0.075 0.011175 0.01186875 0.0125625 0.01325625 0.01395 0.01464375 0.0153375 0.08 0.012288 0.013024 0.01376 0.014496 0.015232 0.015968 0.016704 0.085 0.013447 0.01422475 0.0150025 0.01578025 0.016558 0.01733575 0.0181135 0.09 0.014652 0.015471 0.01629 0.017109 0.017928 0.018747 0.019566 0.095 0.015903 0.01676275 0.0176225 0.01848225 0.019342 0.02020175 0.0210615 0.1 0.0172 0.0181 0.019 0.0199 0.0208 0.0217 0.0226

Table 3:   from 0.075 to 0.105

 0.075 0.080 0.085 0.090 0.095 0.100 0.105 0.005 0.00077125 0.000821 0.00087075 0.0009205 0.00097025 0.00102 0.00106975 0.01 0.001585 0.001684 0.001783 0.001882 0.001981 0.00208 0.002179 0.015 0.00244125 0.002589 0.00273675 0.0028845 0.00303225 0.00318 0.00332775 0.02 0.00334 0.003536 0.003732 0.003928 0.004124 0.00432 0.004516 0.025 0.00428125 0.004525 0.00476875 0.0050125 0.00525625 0.0055 0.00574375 0.03 0.005265 0.005556 0.005847 0.006138 0.006429 0.00672 0.007011 0.035 0.00629125 0.006629 0.00696675 0.0073045 0.00764225 0.00798 0.00831775 0.04 0.00736 0.007744 0.008128 0.008512 0.008896 0.00928 0.009664 0.045 0.00847125 0.008901 0.00933075 0.0097605 0.01019025 0.01062 0.01104975 0.05 0.009625 0.0101 0.010575 0.01105 0.011525 0.012 0.012475 0.055 0.01082125 0.011341 0.01186075 0.0123805 0.01290025 0.01342 0.01393975 0.06 0.01206 0.012624 0.013188 0.013752 0.014316 0.01488 0.015444 0.065 0.01334125 0.013949 0.01455675 0.0151645 0.01577225 0.01638 0.01698775 0.07 0.014665 0.015316 0.015967 0.016618 0.017269 0.01792 0.018571 0.075 0.01603125 0.016725 0.01741875 0.0181125 0.01880625 0.0195 0.02019375 0.08 0.01744 0.018176 0.018912 0.019648 0.020384 0.02112 0.021856 0.085 0.01889125 0.019669 0.02044675 0.0212245 0.02200225 0.02278 0.02355775 0.09 0.020385 0.021204 0.022023 0.022842 0.023661 0.02448 0.025299 0.095 0.02192125 0.022781 0.02364075 0.0245005 0.02536025 0.02622 0.02707975 0.1 0.0235 0.0244 0.0253 0.0262 0.0271 0.028 0.0289

Example of using the tables:

Find the probability of at least two cherries occurring on a payline of a 3-reel slot machine having 65, 65, and 68 stops on reels 1, 2 and 3 respectively and 2, 2, and 1 cherries on these reels respectively.

We have here the particular case of two similar reels (1 and 2).  and . We consider Table 1 and look at the intersection of column  with row , where we find the probability 0.002646. We can take 0.0025 = 0.25% as an approximation of the sought probability. For the exact result, apply directly the explicit probability formula that returned the tables.