3-reel slot machines

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

2.1  Event  – A specific symbol three times

The probability of  is , where  are the basic probabilities of that symbol occurring on the reels respectively.

A particular case is when two of the three reels have the same number of stops and the same distribution of symbols. In that case, 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 three times on a payline

from 0.005 to 0.035

 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.005 0.000000125 0.00000025 0.000000375 0.0000005 0.000000625 0.00000075 0.000000875 0.01 0.0000005 0.000001 0.0000015 0.000002 0.0000025 0.000003 0.0000035 0.015 0.000001125 0.00000225 0.000003375 0.0000045 0.000005625 0.00000675 0.000007875 0.02 0.000002 0.000004 0.000006 0.000008 0.00001 0.000012 0.000014 0.025 0.000003125 0.00000625 0.000009375 0.0000125 0.000015625 0.00001875 0.000021875 0.03 0.0000045 0.000009 0.0000135 0.000018 0.0000225 0.000027 0.0000315 0.035 0.000006125 0.00001225 0.000018375 0.0000245 0.000030625 0.00003675 0.000042875 0.04 0.000008 0.000016 0.000024 0.000032 0.00004 0.000048 0.000056 0.045 0.000010125 0.00002025 0.000030375 0.0000405 0.000050625 0.00006075 0.000070875 0.05 0.0000125 0.000025 0.0000375 0.00005 0.0000625 0.000075 0.0000875 0.055 0.000015125 0.00003025 0.000045375 0.0000605 0.000075625 0.00009075 0.000105875 0.06 0.000018 0.000036 0.000054 0.000072 0.00009 0.000108 0.000126 0.065 0.000021125 0.00004225 0.000063375 0.0000845 0.000105625 0.00012675 0.000147875 0.07 0.0000245 0.000049 0.0000735 0.000098 0.0001225 0.000147 0.0001715 0.075 0.000028125 0.00005625 0.000084375 0.0001125 0.000140625 0.00016875 0.000196875 0.08 0.000032 0.000064 0.000096 0.000128 0.00016 0.000192 0.000224 0.085 0.000036125 0.00007225 0.000108375 0.0001445 0.000180625 0.00021675 0.000252875 0.09 0.0000405 0.000081 0.0001215 0.000162 0.0002025 0.000243 0.0002835 0.095 0.000045125 0.00009025 0.000135375 0.0001805 0.000225625 0.00027075 0.000315875 0.1 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035

Table 2:   from 0.040 to 0.070

 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.005 0.000001 0.000001125 0.00000125 0.000001375 0.0000015 0.000001625 0.00000175 0.01 0.000004 0.0000045 0.000005 0.0000055 0.000006 0.0000065 0.000007 0.015 0.000009 0.000010125 0.00001125 0.000012375 0.0000135 0.000014625 0.00001575 0.02 0.000016 0.000018 0.00002 0.000022 0.000024 0.000026 0.000028 0.025 0.000025 0.000028125 0.00003125 0.000034375 0.0000375 0.000040625 0.00004375 0.03 0.000036 0.0000405 0.000045 0.0000495 0.000054 0.0000585 0.000063 0.035 0.000049 0.000055125 0.00006125 0.000067375 0.0000735 0.000079625 0.00008575 0.04 0.000064 0.000072 0.00008 0.000088 0.000096 0.000104 0.000112 0.045 0.000081 0.000091125 0.00010125 0.000111375 0.0001215 0.000131625 0.00014175 0.05 0.0001 0.0001125 0.000125 0.0001375 0.00015 0.0001625 0.000175 0.055 0.000121 0.000136125 0.00015125 0.000166375 0.0001815 0.000196625 0.00021175 0.06 0.000144 0.000162 0.00018 0.000198 0.000216 0.000234 0.000252 0.065 0.000169 0.000190125 0.00021125 0.000232375 0.0002535 0.000274625 0.00029575 0.07 0.000196 0.0002205 0.000245 0.0002695 0.000294 0.0003185 0.000343 0.075 0.000225 0.000253125 0.00028125 0.000309375 0.0003375 0.000365625 0.00039375 0.08 0.000256 0.000288 0.00032 0.000352 0.000384 0.000416 0.000448 0.085 0.000289 0.000325125 0.00036125 0.000397375 0.0004335 0.000469625 0.00050575 0.09 0.000324 0.0003645 0.000405 0.0004455 0.000486 0.0005265 0.000567 0.095 0.000361 0.000406125 0.00045125 0.000496375 0.0005415 0.000586625 0.00063175 0.1 0.0004 0.00045 0.0005 0.00055 0.0006 0.00065 0.0007

Table 3:   from 0.075 to 0.105

 0.075 0.080 0.085 0.090 0.095 0.100 0.105 0.005 0.000001875 0.000002 0.000002125 0.00000225 0.000002375 0.0000025 0.000002625 0.01 0.0000075 0.000008 0.0000085 0.000009 0.0000095 0.00001 0.0000105 0.015 0.000016875 0.000018 0.000019125 0.00002025 0.000021375 0.0000225 0.000023625 0.02 0.00003 0.000032 0.000034 0.000036 0.000038 0.00004 0.000042 0.025 0.000046875 0.00005 0.000053125 0.00005625 0.000059375 0.0000625 0.000065625 0.03 0.0000675 0.000072 0.0000765 0.000081 0.0000855 0.00009 0.0000945 0.035 0.000091875 0.000098 0.000104125 0.00011025 0.000116375 0.0001225 0.000128625 0.04 0.00012 0.000128 0.000136 0.000144 0.000152 0.00016 0.000168 0.045 0.000151875 0.000162 0.000172125 0.00018225 0.000192375 0.0002025 0.000212625 0.05 0.0001875 0.0002 0.0002125 0.000225 0.0002375 0.00025 0.0002625 0.055 0.000226875 0.000242 0.000257125 0.00027225 0.000287375 0.0003025 0.000317625 0.06 0.00027 0.000288 0.000306 0.000324 0.000342 0.00036 0.000378 0.065 0.000316875 0.000338 0.000359125 0.00038025 0.000401375 0.0004225 0.000443625 0.07 0.0003675 0.000392 0.0004165 0.000441 0.0004655 0.00049 0.0005145 0.075 0.000421875 0.00045 0.000478125 0.00050625 0.000534375 0.0005625 0.000590625 0.08 0.00048 0.000512 0.000544 0.000576 0.000608 0.00064 0.000672 0.085 0.000541875 0.000578 0.000614125 0.00065025 0.000686375 0.0007225 0.000758625 0.09 0.0006075 0.000648 0.0006885 0.000729 0.0007695 0.00081 0.0008505 0.095 0.000676875 0.000722 0.000767125 0.00081225 0.000857375 0.0009025 0.000947625 0.1 0.00075 0.0008 0.00085 0.0009 0.00095 0.001 0.00105

Example of using the tables:

Find the probability of three lemons occurring on a payline of a 3-reel slot machine having 90, 90, and 98 stops on reels 1, 2 and 3 respectively and 3, 3, and 2 lemons on these reels respectively.

We have here the particular case of the same number of stops and the same distribution of the lemon symbol on two reels.              and . We consider Table 2 and look at the intersection of column  with rows  and . We find the probabilities 0.000036 and 0.000049 that bound the sought probability. We can take for instance 0.000040 = 0.004% as an approximation. For the exact result, apply directly the explicit probability formula that returned the tables.