3-reel slot machines

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

2.3  Event  – A specific symbol exactly twice

The probability of  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   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 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 0.000074625 0.00012425 0.000173875 0.0002235 0.000273125 0.00032275 0.000372375 0.01 0.0001985 0.000297 0.0003955 0.000494 0.0005925 0.000691 0.0007895 0.015 0.000371625 0.00051825 0.000664875 0.0008115 0.000958125 0.00110475 0.001251375 0.02 0.000594 0.000788 0.000982 0.001176 0.00137 0.001564 0.001758 0.025 0.000865625 0.00110625 0.001346875 0.0015875 0.001828125 0.00206875 0.002309375 0.03 0.0011865 0.001473 0.0017595 0.002046 0.0023325 0.002619 0.0029055 0.035 0.001556625 0.00188825 0.002219875 0.0025515 0.002883125 0.00321475 0.003546375 0.04 0.001976 0.002352 0.002728 0.003104 0.00348 0.003856 0.004232 0.045 0.002444625 0.00286425 0.003283875 0.0037035 0.004123125 0.00454275 0.004962375 0.05 0.0029625 0.003425 0.0038875 0.00435 0.0048125 0.005275 0.0057375 0.055 0.003529625 0.00403425 0.004538875 0.0050435 0.005548125 0.00605275 0.006557375 0.06 0.004146 0.004692 0.005238 0.005784 0.00633 0.006876 0.007422 0.065 0.004811625 0.00539825 0.005984875 0.0065715 0.007158125 0.00774475 0.008331375 0.07 0.0055265 0.006153 0.0067795 0.007406 0.0080325 0.008659 0.0092855 0.075 0.006290625 0.00695625 0.007621875 0.0082875 0.008953125 0.00961875 0.010284375 0.08 0.007104 0.007808 0.008512 0.009216 0.00992 0.010624 0.011328 0.085 0.007966625 0.00870825 0.009449875 0.0101915 0.010933125 0.01167475 0.012416375 0.09 0.0088785 0.009657 0.0104355 0.011214 0.0119925 0.012771 0.0135495 0.095 0.009839625 0.01065425 0.011468875 0.0122835 0.013098125 0.01391275 0.014727375 0.1 0.01085 0.0117 0.01255 0.0134 0.01425 0.0151 0.01595

Table 2:   from 0.040 to 0.070

 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.005 0.000422 0.000471625 0.00052125 0.000570875 0.0006205 0.000670125 0.00071975 0.01 0.000888 0.0009865 0.001085 0.0011835 0.001282 0.0013805 0.001479 0.015 0.001398 0.001544625 0.00169125 0.001837875 0.0019845 0.002131125 0.00227775 0.02 0.001952 0.002146 0.00234 0.002534 0.002728 0.002922 0.003116 0.025 0.00255 0.002790625 0.00303125 0.003271875 0.0035125 0.003753125 0.00399375 0.03 0.003192 0.0034785 0.003765 0.0040515 0.004338 0.0046245 0.004911 0.035 0.003878 0.004209625 0.00454125 0.004872875 0.0052045 0.005536125 0.00586775 0.04 0.004608 0.004984 0.00536 0.005736 0.006112 0.006488 0.006864 0.045 0.005382 0.005801625 0.00622125 0.006640875 0.0070605 0.007480125 0.00789975 0.05 0.0062 0.0066625 0.007125 0.0075875 0.00805 0.0085125 0.008975 0.055 0.007062 0.007566625 0.00807125 0.008575875 0.0090805 0.009585125 0.01008975 0.06 0.007968 0.008514 0.00906 0.009606 0.010152 0.010698 0.011244 0.065 0.008918 0.009504625 0.01009125 0.010677875 0.0112645 0.011851125 0.01243775 0.07 0.009912 0.0105385 0.011165 0.0117915 0.012418 0.0130445 0.013671 0.075 0.01095 0.011615625 0.01228125 0.012946875 0.0136125 0.014278125 0.01494375 0.08 0.012032 0.012736 0.01344 0.014144 0.014848 0.015552 0.016256 0.085 0.013158 0.013899625 0.01464125 0.015382875 0.0161245 0.016866125 0.01760775 0.09 0.014328 0.0151065 0.015885 0.0166635 0.017442 0.0182205 0.018999 0.095 0.015542 0.016356625 0.01717125 0.017985875 0.0188005 0.019615125 0.02042975 0.1 0.0168 0.01765 0.0185 0.01935 0.0202 0.02105 0.0219

Table 3:   from 0.075 to 0.105

 0.075 0.080 0.085 0.090 0.095 0.100 0.105 0.005 0.000769375 0.000819 0.000868625 0.00091825 0.000967875 0.0010175 0.001067125 0.01 0.0015775 0.001676 0.0017745 0.001873 0.0019715 0.00207 0.0021685 0.015 0.002424375 0.002571 0.002717625 0.00286425 0.003010875 0.0031575 0.003304125 0.02 0.00331 0.003504 0.003698 0.003892 0.004086 0.00428 0.004474 0.025 0.004234375 0.004475 0.004715625 0.00495625 0.005196875 0.0054375 0.005678125 0.03 0.0051975 0.005484 0.0057705 0.006057 0.0063435 0.00663 0.0069165 0.035 0.006199375 0.006531 0.006862625 0.00719425 0.007525875 0.0078575 0.008189125 0.04 0.00724 0.007616 0.007992 0.008368 0.008744 0.00912 0.009496 0.045 0.008319375 0.008739 0.009158625 0.00957825 0.009997875 0.0104175 0.010837125 0.05 0.0094375 0.0099 0.0103625 0.010825 0.0112875 0.01175 0.0122125 0.055 0.010594375 0.011099 0.011603625 0.01210825 0.012612875 0.0131175 0.013622125 0.06 0.01179 0.012336 0.012882 0.013428 0.013974 0.01452 0.015066 0.065 0.013024375 0.013611 0.014197625 0.01478425 0.015370875 0.0159575 0.016544125 0.07 0.0142975 0.014924 0.0155505 0.016177 0.0168035 0.01743 0.0180565 0.075 0.015609375 0.016275 0.016940625 0.01760625 0.018271875 0.0189375 0.019603125 0.08 0.01696 0.017664 0.018368 0.019072 0.019776 0.02048 0.021184 0.085 0.018349375 0.019091 0.019832625 0.02057425 0.021315875 0.0220575 0.022799125 0.09 0.0197775 0.020556 0.0213345 0.022113 0.0228915 0.02367 0.0244485 0.095 0.021244375 0.022059 0.022873625 0.02368825 0.024502875 0.0253175 0.026132125 0.1 0.02275 0.0236 0.02445 0.0253 0.02615 0.027 0.02785

Example of using the tables:

Find the probability of two cherries occurring on a payline of a 3-reel slot machine having 72, 72, and 72 stops on reels 1, 2, and 3 respectively and 4, 2, and 2 cherries on these reels respectively.

We have here the particular case of the same number of stops and the same distribution of the cherry symbol on two reels (2 and 3).              and . We consider Table 2 and look at the intersection of column  with rows  and . We find the probabilities 0.003271875 and 0.0040515 that bound the sought probability. We can take for instance 0.0036 = 0.36% as an approximation. For the exact result, apply directly the explicit probability formula that returned the tables.