Calculate Probability Of Slot Machine
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- For the school fayre we are planning to make a slot machine with three wheels and three images on each wheel. What are the odds or what is the probability of a 'win'? Obviously we don't want to make the chance of getting three identical images as likely as a Vegas jackpot, but it would be nice to understand the maths behind such things - just.
- Early slot machines were mechanical devices. They had three metal reels that had ten possible stops each. To calculate the odds of a single symbol appearing on a reel, you just divide the one symbol by the total number of potential outcomes. So if you had one cherry on a reel, your odds of hitting that cherry were 1/10, or 10%.
- The Computerized Slot Machines are harder to calculate because they use complex computer programs to generate the random numbers for which symbol appears on the machine. I won’t be able to give you a calculation, but I will try to explain how casinos use probability to ruin our chances of winning.
After many years of calculating in my head how often a slot machine pays out, I finally decided to create a simple app to track it. The purpose of this app is to determine the payout probability of a slot machine. See the Instructions screen in-app for more details.
First of all, we must start with the number of possible combinations. In the case of slots, it is relatively simple – just multiply the numbers of symbols on each reel. The oldest slots had, for example, 3 reels with ten different symbols on each. The total number of combinations that could appear on the panel was 1,000 (10 x 10 x 10).
The number of combinations in today’s slots is somewhat higher. If we assume five reels with 30 symbols on each, we get a total of 243,000,000 combinations.
If you want to calculate your chances to win on an online slot machine, all you need is this simple equation:
Number of winning combinations / Total number of combinations
To calculate the payout of the slot machine, modify the formula a little:
Σ (winning combination_k * possible yield_k) / (Total number of combinations)
Let’s analyze a few basic slot machines. For the purposes of our article and in order to simplify the calculation, we will assume that the slot machine has only one payout line and the bet is one coin per round.
Analysis of the simplest slot machine
Let’s go back to the past and assume that the machine only has 3 reels and there is an apple, an orange, a lemon, a banana, a melon and a joker symbol on each. The individual combinations produce these winnings:
- Three jokers win 30 coins
- Any three fruits win 10 coins
- Two jokers win 4 coins
- One joker wins 1 coin
The total number of combinations is 216 (6 x 6 x 6).
Total number of winning combinations:
- In the first case there is only one winning combination (1 x 1 x 1 = 1)
- In the second case we have 5 winning combinations (3 times apple or 3 times orange or 3 times lemon, …) (1 x 1 x 1) x 5 = 5
- The joker may appear on any two reels. The calculation is as follows: 1 x 1 x 5 + 1 x 5 x 1 + 5 x 1 x 1 = 15
- The joker may appear on any reel. 1 × 5 × 5 + 5 × 1 × 5 + 5 × 5 × 1 = 75
Our simplified model thus contains 1 + 5 + 15 + 75 = 96 winning combinations. The table below shows the probability of a payout.
| Winning combination | Number of combinations | Winning | Returns for 1 coin | Chance to win |
| 3 jokers | 1 | 30 | 30 | 13.953% |
| Any fruit | 5 | 10 | 50 | 23.256% |
| 2 jokers | 15 | 4 | 60 | 27.907% |
| 1 joker | 75 | 1 | 75 | 34.884% |
| Total | 96 | 215 | ||
| % for the winning combination | 44.444% | Payouts | 99.537% |
Calculation of payouts according to the formula
Σ (winning combination_k * possible yield_k) / (Total number of combinations)
(1 × 30 + 5 × 50 + 15 × 4 + 75 × 1)/(6 × 6 × 6) = 215/216 ≈ 0.99537
In this case, the slot machine has a payout ratio of 99.53%, which is very nice, but in a real casino, you will not find the same results. The average returns of slots online casinos will be between 94% and 98%.
The table also clearly shows how single coin wins affect payouts. If the win of each combination were equal to one coin, the winning ratio would drop to 44.4%. And that’s a very small number.
Analysis of a more complicated slot
Because the previous example was too distant from reality, let’s show you another example with higher numbers. To simplify, let’s assume again that there is only one payline, the slot machine has 3 reels and a total of 6 symbols that can appear on the panel:
| Symbol | Reel 1 | Reel 2 | Reel 3 |
| BAR | 1 | 1 | 1 |
| SEVEN | 3 | 1 | 1 |
| Cherry | 4 | 3 | 3 |
| Orange | 5 | 6 | 6 |
| Banana | 5 | 6 | 6 |
| Lemon | 5 | 6 | 6 |
| Total | 23 | 23 | 23 |
The total number of combinations is 23 x 23 x 23 = 12,167.
Winning combinations with single coin returns:
- 3x BAR, win 60 coins, number of combinations 1
- 3x SEVEN, win 40 coins, number of combinations 3 x 1 x 1 = 3
- 3x Cherry, win 20 coins, number of combinations 4 x 3 x 3 = 36
- 3x Other fruit, win 10 coins, number of combinations (5 x 6 x 6) x 3 = 540
- Cherry on two reels, win 4 coins, number of combinations 651
- Cherry on one reel, win 1 coin, number of winning combinations 3,880
Calculation for no. 5:
Cherry, Cherry, Other: 4 x 3 x (23 – 3) = 240
Cherry, Other, Cherry: 4 x (23 – 3) x 3 = 240
Other, Cherry, Cherry: (23 – 4) x 3 x 3 = 171
Calculation for no. 6:
First reel: 4 x 20 x 20 = 1,600
Second reel 19 x 3 x 20 = 1,120
Calculate Probability Of Slot Machines
Third reel 19 x 20 x 3 = 1,120
The following table shows the amount of payout and the chance of winning for the individual combinations.
| Winning combination | Number of combinations | Winning | Returns for 1 coin | Chance to win |
| 3x BAR | 1 | 60 | 60 | 0.495% |
| 3x SEVEN | 3 | 40 | 120 | 0.989% |
| 3x Cherry | 36 | 20 | 720 | 5.934% |
| 3x Other fruit | 540 | 10 | 5,400 | 44.507% |
| 2x Cherry | 651 | 3 | 1,935 | 16.097% |
| 1x Cherry | 3,880 | 1 | 3,880 | 31.979% |
| Total | 5,111 | 12,133 | ||
| % of winning combinations | 42.007% | Payout | 99.721% |
As you can see, the payout ratio is very high again at 99.721% (12,133 / 12,161). If the slot were to pay a straight win for each winning combination in the amount of 1 coin, the payout ratio would be down to 42,007%.
Calculate Probability Of Slot Machines
Random chance plays an important role of every type of gambling game. Skill is the leverage (in terms of knowledge of the rules of the game) you apply to maximize your potential for winning andminimize your potential for losing. No casino gambling game is ever determined by skill alone.
We cannot measure random chance. And gamblers frequently mistake their knowledge of games’ probability tables for predictive knowledge. Probabilities do not predict anything. If you know the probabilities for the outcome of the next round in a casinogame, you can feel more confident about your next action. That is all the probabilities guarantee you.
The casino, on the other hand, uses the probabilities to determine the maximum risk itaccepts per round in a game. The casino represents those acceptable risks in terms of the odds they give per wager.
The mathematical probability of an event happening is 1 divided by the number of possible events. You add the probabilities for identical events together to compute unique probabilities. Butbecause random chance is used to determine game outcomes, the probabilities have no effect on what happens next. And the odds do not reflect the probabilities.