Slot Machine Expected Value
Posted : admin On 4/7/2022
- 3-reel slot machine basic probabilities calculations Cartesian grid cherry occurring combinations of stops combinations of symbols complex events configuration defined Definition denoted diagonal lines display distinct symbols distribution of symbols elementary events estimation event E Event E3 events of type events related Example expected.
- 3b: Slot Machine Return Percentages in Jean & Primm, Nevada 3c: Slot Machine returns in Tunica, MS (Mississippi) 3d: Slot Machine Return Percentages for Henderson/Lake Mead Casinos.
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- Appendices
- Slots Analysis
- Miscellaneous
One of the great things about video poker is that even though it looks like a slot machine, it’s NOT a slot machine. Here’s why that’s great: Slot machines are the only game in the casino that have opaque math behind them. In other words, you have no way of knowing what the odds and/or payouts are on a slots game. Yes, slots have pay tables. A slot machine house edge is known by casino managers as the “hold”, and hold percentages vary a great deal, and do tend to be smaller at more expensive slots, frequently found to be around 1- 3% at the five dollar slots.
Introduction
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:
Slot Machine
| 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
Slot Machine Expected Values
| 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 pay table 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.