–
A specific symbol three times3-reel slot machines
1. Case A – the same number of stops and symbol distribution on the reels
1.1 Event
–
A specific symbol three times
The probability of
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. The values of t are listed in increments of 2, ranging from 20 to
200, and the values of c range from 1 to 7.
Table of values for the probability of a specific symbol occurring three times on a payline
|
c |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
|
t |
|||||||
|
20 |
0.000125 |
0.001 |
0.003375 |
0.008 |
0.015625 |
0.027 |
0.042875 |
|
22 |
9.3914E-05 |
0.00075131 |
0.00253569 |
0.00601052 |
0.01173929 |
0.0202855 |
0.03221262 |
|
24 |
7.2338E-05 |
0.0005787 |
0.00195313 |
0.00462963 |
0.00904225 |
0.015625 |
0.02481192 |
|
26 |
5.6896E-05 |
0.00045517 |
0.00153619 |
0.00364133 |
0.00711197 |
0.01228949 |
0.01951525 |
|
28 |
4.5554E-05 |
0.00036443 |
0.00122996 |
0.00291545 |
0.00569424 |
0.00983965 |
0.015625 |
|
30 |
3.7037E-05 |
0.0002963 |
0.001 |
0.00237037 |
0.00462963 |
0.008 |
0.0127037 |
|
32 |
3.0518E-05 |
0.00024414 |
0.00082397 |
0.00195313 |
0.0038147 |
0.0065918 |
0.01046753 |
|
34 |
2.5443E-05 |
0.00020354 |
0.00068695 |
0.00162833 |
0.00318034 |
0.00549562 |
0.00872685 |
|
36 |
2.1433E-05 |
0.00017147 |
0.0005787 |
0.00137174 |
0.00267918 |
0.00462963 |
0.00735168 |
|
38 |
1.8224E-05 |
0.00014579 |
0.00049205 |
0.00116635 |
0.00227803 |
0.00393643 |
0.00625091 |
|
40 |
1.5625E-05 |
0.000125 |
0.00042188 |
0.001 |
0.00195313 |
0.003375 |
0.00535938 |
|
42 |
1.3497E-05 |
0.00010798 |
0.00036443 |
0.00086384 |
0.00168718 |
0.00291545 |
0.00462963 |
|
44 |
1.1739E-05 |
9.3914E-05 |
0.00031696 |
0.00075131 |
0.00146741 |
0.00253569 |
0.00402658 |
|
46 |
1.0274E-05 |
8.219E-05 |
0.00027739 |
0.00065752 |
0.00128421 |
0.00221912 |
0.00352388 |
|
48 |
9.0422E-06 |
7.2338E-05 |
0.00024414 |
0.0005787 |
0.00113028 |
0.00195313 |
0.00310149 |
|
50 |
0.000008 |
0.000064 |
0.000216 |
0.000512 |
0.001 |
0.001728 |
0.002744 |
|
52 |
7.112E-06 |
5.6896E-05 |
0.00019202 |
0.00045517 |
0.000889 |
0.00153619 |
0.00243941 |
|
54 |
6.3507E-06 |
5.0805E-05 |
0.00017147 |
0.00040644 |
0.00079383 |
0.00137174 |
0.00217828 |
|
56 |
5.6942E-06 |
4.5554E-05 |
0.00015374 |
0.00036443 |
0.00071178 |
0.00122996 |
0.00195313 |
|
58 |
5.1253E-06 |
4.1002E-05 |
0.00013838 |
0.00032802 |
0.00064066 |
0.00110706 |
0.00175796 |
|
60 |
4.6296E-06 |
3.7037E-05 |
0.000125 |
0.0002963 |
0.0005787 |
0.001 |
0.00158796 |
|
62 |
4.1959E-06 |
3.3567E-05 |
0.00011329 |
0.00026854 |
0.00052449 |
0.00090631 |
0.00143919 |
|
64 |
3.8147E-06 |
3.0518E-05 |
0.000103 |
0.00024414 |
0.00047684 |
0.00082397 |
0.00130844 |
|
66 |
3.4783E-06 |
2.7826E-05 |
9.3914E-05 |
0.00022261 |
0.00043479 |
0.00075131 |
0.00119306 |
|
68 |
3.1803E-06 |
2.5443E-05 |
8.5869E-05 |
0.00020354 |
0.00039754 |
0.00068695 |
0.00109086 |
|
70 |
2.9155E-06 |
2.3324E-05 |
7.8717E-05 |
0.00018659 |
0.00036443 |
0.00062974 |
0.001 |
|
72 |
2.6792E-06 |
2.1433E-05 |
7.2338E-05 |
0.00017147 |
0.0003349 |
0.0005787 |
0.00091896 |
|
74 |
2.4678E-06 |
1.9742E-05 |
6.663E-05 |
0.00015794 |
0.00030847 |
0.00053304 |
0.00084645 |
|
76 |
2.278E-06 |
1.8224E-05 |
6.1507E-05 |
0.00014579 |
0.00028475 |
0.00049205 |
0.00078136 |
|
78 |
2.1073E-06 |
1.6858E-05 |
5.6896E-05 |
0.00013486 |
0.00026341 |
0.00045517 |
0.00072279 |
|
80 |
1.9531E-06 |
1.5625E-05 |
5.2734E-05 |
0.000125 |
0.00024414 |
0.00042188 |
0.00066992 |
|
82 |
1.8137E-06 |
1.4509E-05 |
4.8969E-05 |
0.00011607 |
0.00022671 |
0.00039175 |
0.00062209 |
|
84 |
1.6872E-06 |
1.3497E-05 |
4.5554E-05 |
0.00010798 |
0.0002109 |
0.00036443 |
0.0005787 |
|
86 |
1.5722E-06 |
1.2578E-05 |
4.2449E-05 |
0.00010062 |
0.00019652 |
0.00033959 |
0.00053926 |
|
88 |
1.4674E-06 |
1.1739E-05 |
3.962E-05 |
9.3914E-05 |
0.00018343 |
0.00031696 |
0.00050332 |
|
90 |
1.3717E-06 |
1.0974E-05 |
3.7037E-05 |
8.7791E-05 |
0.00017147 |
0.0002963 |
0.00047051 |
|
92 |
1.2842E-06 |
1.0274E-05 |
3.4674E-05 |
8.219E-05 |
0.00016053 |
0.00027739 |
0.00044048 |
|
94 |
1.204E-06 |
9.6318E-06 |
3.2507E-05 |
7.7054E-05 |
0.0001505 |
0.00026006 |
0.00041296 |
|
96 |
1.1303E-06 |
9.0422E-06 |
3.0518E-05 |
7.2338E-05 |
0.00014129 |
0.00024414 |
0.00038769 |
|
98 |
1.0625E-06 |
8.4999E-06 |
2.8687E-05 |
6.7999E-05 |
0.00013281 |
0.0002295 |
0.00036443 |
|
100 |
0.000001 |
0.000008 |
0.000027 |
0.000064 |
0.000125 |
0.000216 |
0.000343 |
|
102 |
9.4232E-07 |
7.5386E-06 |
2.5443E-05 |
6.0309E-05 |
0.00011779 |
0.00020354 |
0.00032322 |
|
104 |
8.89E-07 |
7.112E-06 |
2.4003E-05 |
5.6896E-05 |
0.00011112 |
0.00019202 |
0.00030493 |
|
106 |
8.3962E-07 |
6.717E-06 |
2.267E-05 |
5.3736E-05 |
0.00010495 |
0.00018136 |
0.00028799 |
|
108 |
7.9383E-07 |
6.3507E-06 |
2.1433E-05 |
5.0805E-05 |
9.9229E-05 |
0.00017147 |
0.00027228 |
|
110 |
7.5131E-07 |
6.0105E-06 |
2.0285E-05 |
4.8084E-05 |
9.3914E-05 |
0.00016228 |
0.0002577 |
|
112 |
7.1178E-07 |
5.6942E-06 |
1.9218E-05 |
4.5554E-05 |
8.8973E-05 |
0.00015374 |
0.00024414 |
|
114 |
6.7497E-07 |
5.3998E-06 |
1.8224E-05 |
4.3198E-05 |
8.4371E-05 |
0.00014579 |
0.00023152 |
|
116 |
6.4066E-07 |
5.1253E-06 |
1.7298E-05 |
4.1002E-05 |
8.0082E-05 |
0.00013838 |
0.00021975 |
|
118 |
6.0863E-07 |
4.869E-06 |
1.6433E-05 |
3.8952E-05 |
7.6079E-05 |
0.00013146 |
0.00020876 |
|
120 |
5.787E-07 |
4.6296E-06 |
1.5625E-05 |
3.7037E-05 |
7.2338E-05 |
0.000125 |
0.0001985 |
|
122 |
5.5071E-07 |
4.4057E-06 |
1.4869E-05 |
3.5245E-05 |
6.8838E-05 |
0.00011895 |
0.00018889 |
|
124 |
5.2449E-07 |
4.1959E-06 |
1.4161E-05 |
3.3567E-05 |
6.5561E-05 |
0.00011329 |
0.0001799 |
|
126 |
4.9991E-07 |
3.9992E-06 |
1.3497E-05 |
3.1994E-05 |
6.2488E-05 |
0.00010798 |
0.00017147 |
|
128 |
4.7684E-07 |
3.8147E-06 |
1.2875E-05 |
3.0518E-05 |
5.9605E-05 |
0.000103 |
0.00016356 |
|
130 |
4.5517E-07 |
3.6413E-06 |
1.2289E-05 |
2.9131E-05 |
5.6896E-05 |
9.8316E-05 |
0.00015612 |
|
132 |
4.3479E-07 |
3.4783E-06 |
1.1739E-05 |
2.7826E-05 |
5.4349E-05 |
9.3914E-05 |
0.00014913 |
|
134 |
4.1561E-07 |
3.3249E-06 |
1.1221E-05 |
2.6599E-05 |
5.1951E-05 |
8.9772E-05 |
0.00014255 |
|
136 |
3.9754E-07 |
3.1803E-06 |
1.0734E-05 |
2.5443E-05 |
4.9693E-05 |
8.5869E-05 |
0.00013636 |
|
138 |
3.8051E-07 |
3.0441E-06 |
1.0274E-05 |
2.4352E-05 |
4.7563E-05 |
8.219E-05 |
0.00013051 |
|
140 |
3.6443E-07 |
2.9155E-06 |
9.8397E-06 |
2.3324E-05 |
4.5554E-05 |
7.8717E-05 |
0.000125 |
|
142 |
3.4925E-07 |
2.794E-06 |
9.4297E-06 |
2.2352E-05 |
4.3656E-05 |
7.5438E-05 |
0.00011979 |
|
144 |
3.349E-07 |
2.6792E-06 |
9.0422E-06 |
2.1433E-05 |
4.1862E-05 |
7.2338E-05 |
0.00011487 |
|
148 |
3.0847E-07 |
2.4678E-06 |
8.3287E-06 |
1.9742E-05 |
3.8559E-05 |
6.663E-05 |
0.00010581 |
|
150 |
2.963E-07 |
2.3704E-06 |
0.000008 |
1.8963E-05 |
3.7037E-05 |
0.000064 |
0.00010163 |
|
152 |
2.8475E-07 |
2.278E-06 |
7.6883E-06 |
1.8224E-05 |
3.5594E-05 |
6.1507E-05 |
9.767E-05 |
|
154 |
2.738E-07 |
2.1904E-06 |
7.3927E-06 |
1.7523E-05 |
3.4225E-05 |
5.9141E-05 |
9.3914E-05 |
|
156 |
2.6341E-07 |
2.1073E-06 |
7.112E-06 |
1.6858E-05 |
3.2926E-05 |
5.6896E-05 |
9.0348E-05 |
|
158 |
2.5353E-07 |
2.0282E-06 |
6.8453E-06 |
1.6226E-05 |
3.1691E-05 |
5.4762E-05 |
8.6961E-05 |
|
160 |
2.4414E-07 |
1.9531E-06 |
6.5918E-06 |
1.5625E-05 |
3.0518E-05 |
5.2734E-05 |
8.374E-05 |
|
162 |
2.3521E-07 |
1.8817E-06 |
6.3507E-06 |
1.5053E-05 |
2.9401E-05 |
5.0805E-05 |
8.0677E-05 |
|
164 |
2.2671E-07 |
1.8137E-06 |
6.1211E-06 |
1.4509E-05 |
2.8339E-05 |
4.8969E-05 |
7.7761E-05 |
|
166 |
2.1861E-07 |
1.7489E-06 |
5.9025E-06 |
1.3991E-05 |
2.7327E-05 |
4.722E-05 |
7.4984E-05 |
|
168 |
2.109E-07 |
1.6872E-06 |
5.6942E-06 |
1.3497E-05 |
2.6362E-05 |
4.5554E-05 |
7.2338E-05 |
|
170 |
2.0354E-07 |
1.6283E-06 |
5.4956E-06 |
1.3027E-05 |
2.5443E-05 |
4.3965E-05 |
6.9815E-05 |
|
172 |
1.9652E-07 |
1.5722E-06 |
5.3061E-06 |
1.2578E-05 |
2.4565E-05 |
4.2449E-05 |
6.7408E-05 |
|
174 |
1.8982E-07 |
1.5186E-06 |
5.1253E-06 |
1.2149E-05 |
2.3728E-05 |
4.1002E-05 |
6.511E-05 |
|
176 |
1.8343E-07 |
1.4674E-06 |
4.9525E-06 |
1.1739E-05 |
2.2928E-05 |
3.962E-05 |
6.2915E-05 |
|
178 |
1.7731E-07 |
1.4185E-06 |
4.7874E-06 |
1.1348E-05 |
2.2164E-05 |
3.83E-05 |
6.0818E-05 |
|
180 |
1.7147E-07 |
1.3717E-06 |
4.6296E-06 |
1.0974E-05 |
2.1433E-05 |
3.7037E-05 |
5.8813E-05 |
|
182 |
1.6588E-07 |
1.327E-06 |
4.4787E-06 |
1.0616E-05 |
2.0735E-05 |
3.5829E-05 |
5.6896E-05 |
|
184 |
1.6053E-07 |
1.2842E-06 |
4.3342E-06 |
1.0274E-05 |
2.0066E-05 |
3.4674E-05 |
5.5061E-05 |
|
186 |
1.554E-07 |
1.2432E-06 |
4.1959E-06 |
9.9458E-06 |
1.9425E-05 |
3.3567E-05 |
5.3303E-05 |
|
188 |
1.505E-07 |
1.204E-06 |
4.0634E-06 |
9.6318E-06 |
1.8812E-05 |
3.2507E-05 |
5.162E-05 |
|
190 |
1.4579E-07 |
1.1664E-06 |
3.9364E-06 |
9.3308E-06 |
1.8224E-05 |
3.1491E-05 |
5.0007E-05 |
|
192 |
1.4129E-07 |
1.1303E-06 |
3.8147E-06 |
9.0422E-06 |
1.7661E-05 |
3.0518E-05 |
4.8461E-05 |
|
194 |
1.3696E-07 |
1.0957E-06 |
3.6979E-06 |
8.7655E-06 |
1.712E-05 |
2.9583E-05 |
4.6977E-05 |
|
196 |
1.3281E-07 |
1.0625E-06 |
3.5859E-06 |
8.4999E-06 |
1.6601E-05 |
2.8687E-05 |
4.5554E-05 |
|
198 |
1.2883E-07 |
1.0306E-06 |
3.4783E-06 |
8.2449E-06 |
1.6103E-05 |
2.7826E-05 |
4.4187E-05 |
|
200 |
1.25E-07 |
0.000001 |
3.375E-06 |
0.000008 |
1.5625E-05 |
0.000027 |
4.2875E-05 |
Note: The notation of a number using letter E – called scientific notation – applies for very low numbers with a large number of decimal places. To convert a number from scientific notation to its standard decimal notation, we must move the decimal point to the left the number of decimal places indicated by the number written after “E-”. For instance, 1.25E-07 converts to 0.000000125, which means a 0.0000125% probability.
Examples of using the table:
1) Find the probability of a cherry occurring three times on a payline of a 3-reel slot machine with 82 stops on each reel, having 4 cherries on each reel.
We look at the intersection of row t = 82 with column c = 4 and find the probability P = 0.00011607 = 0.011607%.
2) Find the probability of a red seven occurring three times on a payline of a 3-reel slot machine with 51 stops on each reel, having 5 red sevens on each reel.
We observe that 50 < 51 < 52.
We look at the intersection of row t = 50 with column c = 5 and find the first probability as 0.001. We look at the intersection of row t = 50 with column c = 5 and find the second probability as 0.000889.
The sought probability is between 0.000889 and 0.001. We could choose for instance P = 0.00095 = 0.095% as an approximated probability.