Probability Of A Slot Machine
If there is a slot machine with 4 slots, each with a possibility of being either a cherry, a lemon, or a 7, what is the probability of getting exactly two lemons? My thinking: 4 slot machine, 3 options: 3.3.3.3 = 81 total options. Getting exactly 2 lemons: 1.1.3.3 = 9. P(LL) = 9/81 = 0.1111. What is wrong with this?
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).
- Suppose you have a slot machine with three reels with ten symbols on each, and it only pays out when three cherries hit. The odds of winning that jackpot, as we determined earlier, is 1/1000. If we set the jackpot as $900, and charge $1 per bet, the payout percentage for that game will be 90%, or $900/$1000.
- Slot Machines The probability of winning on a slot machine is 5%. If a person plays the machine 500 times, find the probability of winning 30 times. Use the normal approximation to the binomial distribution. Multiple-Job Holders According to the government 5.3% of those employed are multiple-job holders.
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
Probability Of Winning On A Slot Machine
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:
Calculate Probability Of Slot Machines
First reel: 4 x 20 x 20 = 1,600
Second reel 19 x 3 x 20 = 1,120
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%.