Appendix 2
From time to time I get asked specifically how to
calculate the return for a slot machine. To avoid breaking any
copyright laws I won't use any actual machine as an example but
create me own. Lets assume this is a standard three reel
electro-mechanical slot machine with the following payoff table
based on the center line:
| Center Payline |
Pays |
| Three bars |
5000 |
| Three cherries |
1000 |
| Three plums |
200 |
| Three watermelons |
100 |
| Three oranges |
50 |
| Three lemons |
25 |
| Any two cherries |
10 |
| Any one cherry |
2 |
There seems to be always 22 actual stops on each reel of a slot
machine. The following table shows the symbol on each stop as well
as the weight.
| Weight Table |
| Symbol |
Reel 1 |
Reel 2 |
Reel 3 |
| Cherry |
3 |
2 |
1 |
| Blank |
2 |
3 |
3 |
| Plum |
3 |
2 |
2 |
| Blank |
2 |
3 |
3 |
| Watermelon |
3 |
3 |
2 |
| Blank |
2 |
3 |
3 |
| Orange |
4 |
3 |
3 |
| Blank |
2 |
3 |
3 |
| Lemon |
4 |
3 |
3 |
| Blank |
5 |
5 |
8 |
| Bar |
4 |
3 |
1 |
| Blank |
5 |
5 |
7 |
| Cherry |
2 |
2 |
1 |
| Blank |
2 |
3 |
3 |
| Plum |
3 |
2 |
1 |
| Blank |
2 |
3 |
3 |
| Watermelon |
3 |
2 |
2 |
| Blank |
2 |
3 |
3 |
| Orange |
3 |
2 |
3 |
| Blank |
2 |
3 |
3 |
| Lemon |
4 |
3 |
3 |
| Blank |
2 |
3 |
3 |
| Total |
64 |
64 |
64 |
There are two interesting things to note at this point. First
notice that the first reel is weight the most generously and the
third is the least. For example the bar has 4 weights on reel 1 and
only 1 weight on reel 3. Second notice the high number of blanks
directly above and below the bar symbol. This results in a near miss
effect.
Most of the symbols occur twice on the reel, and the blank 11
times. The following table shows the total number of weights of each
kind of symbol.
| Total Weight Table |
| Symbol |
Reel 1 |
Reel 2 |
Reel 3 |
| Bar |
4 |
3 |
1 |
| Cherry |
5 |
4 |
2 |
| Plum |
6 |
4 |
3 |
| Watermelon |
6 |
5 |
4 |
| Orange |
7 |
5 |
6 |
| Lemon |
8 |
6 |
6 |
| Blank |
28 |
37 |
42 |
| Total |
64 |
64 |
64 |
Given the two table of weights and the paytable it only takes
simple math to calculate the expected return. Following are the
specific probabilities of each paying combination. Note that each
virtual reel has a total of 64 stops so the total number of possible
combinations is 643 = 262,144.
- 3 Bars: 4*3*1/262,144 = 0.000046
- 3 Cherries: 5*4*2/262,144 = 0.000153
- 3 Plums: 6*4*3/262,144 = 0.000275
- 3 Watermelons: 6*5*4/262,144 = 0.000458
- 3 Oranges: 7*5*6/262,144 = 0.000801
- 3 Lemons: 8*6*6/262,144 = 0.001099
- 2 Cherries: (5*4*62 + 5*60*2 + 59*4*2)/262,144 = 0.008820
- 1 Cherry: (5*60*62 + 59*4*62 + 59*60*2)/262,144 = 0.153778
The average return of the machine is the dot product of
the above probabilities and their respective payoffs:
0.000046*5000 + 0.000153*1000 + 0.000275*200 + 0.000458*100 +
0.000801*50 + 0.001099*25 + 0.008820*10 + 0.153778*2 = 0.94545 .
Thus for every unit played the machine will return back 94.545%.
Go black to slot
machines.
