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.

1. Calculate Probability Of Slot Machines
2. Calculate Probability Of Slot Machines

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.

Discussed in this text range from the basic principles of probability, odds, expectation, house advantage, and the law of averages, to price setting using game odds, gaming and economic regulations, standards for fairness, player worth, and rebate programs. I'm trying to calculate the probability to win in a slot machine. Return-to-Player is 96%. I have used the sum of the Binomial Coefficient to calculate it. Since Return to Player is 96%, I make the assumption that the chances to win is 48% given that you're paid 1:1. (Please correct me if I'm wrong).

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:

1. Three jokers win 30 coins
2. Any three fruits win 10 coins
3. Two jokers win 4 coins
4. One joker wins 1 coin

The total number of combinations is 216 (6 x 6 x 6).

Total number of winning combinations:

1. In the first case there is only one winning combination (1 x 1 x 1 = 1)
2. 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
3. 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
4. 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.

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:

The total number of combinations is 23 x 23 x 23 = 12,167.

Winning combinations with single coin returns:

1. 3x BAR, win 60 coins, number of combinations 1
2. 3x SEVEN, win 40 coins, number of combinations 3 x 1 x 1 = 3
3. 3x Cherry, win 20 coins, number of combinations 4 x 3 x 3 = 36
4. 3x Other fruit, win 10 coins, number of combinations (5 x 6 x 6) x 3 = 540
5. Cherry on two reels, win 4 coins, number of combinations 651
6. 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

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.

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%.

If you’re the type of person who isn’t that great at Blackjack or Poker and feel left out when you go with your friends to a casino, you can always spend your time on the slot machines. Slot Machines are those big boxes shaped like vending machines that have bright, flashy lights and play wonky circus music. In a big casino like the ones in Macau or Las Vegas, you will have rooms with rows of slot machines and slot machines in random corners of the casino. They are the perfect gambling entertainment for the rookie gambler and the newbie casino goer. It’s a simple game and I will try to explain how it works and your chances of winning a jackpot.

All slot machines will have the same type of features. They will have:

• A Lever: It’s the big handle that sticks out from the side of the machine. If you pull it, you start the game.
• Reels: The spinning wheels inside the machine that have different types of symbols and pictures on them. They start turning once you pull the lever. Then all the reels start turning and stop one-by-one from left to right. You win the game based on the arrangement of the types of symbols that appear on the reels. There are generally in most slot machines three spinning reels.

There are two types of slot machines that are based on how they are made. There is the classic, mechanical slot machine that turns with the reels with gears and pulleys, and the modern, computerized slot machine that uses a computer program to turn the reels. The computerized slot machines can allow casinos to be really creative, and sneaky, with how they set them up.

Let’s set up a make-believe mechanical slot machine that has three reels with 64 symbols on each. To make things simple, to get a jackpot, you need to have a ‘7’ symbol on each reel, other combinations are losers. The first reel has no effect on where the second reel stops, and the second reel has no effect on where the third reel stops, making each reel independent of the other reels.

• If each reel has 64 symbols, of which only the ‘7’ symbol can let us win, therefore the probability of getting a ‘7’ on the first reel is 1 out of 64.
• Since all the reels are identical, that means the probability of getting a ‘7’ on the second reel is also 1 out of 64.
• Same like before, the probability of getting a ‘7’ on the third reel is also 1 out of 64.
• Since each reel is an independent event, the formula to calculate the probability of getting a ‘7’ on each reel is P(Three ‘7’s) = P(‘7’ on first reel)*P(‘7’ on second reel)*P(‘7’ on third reel)
• Therefore: P(Three ‘7’s or Jackpot) = (1/64)(1/64)(1/64) = 1/262144 or 1/(2^18)
• The odds of getting a jackpot or three ‘7’s in a row are: P(Jackpot)/P(Losing) = P(Jackpot)=P'(Jackpot) = P(Jackpot)=(1-P(Jackpot)) = (1/262144)/(262143/262144) = 1:262143. If you really think about it, it seems almost impossible to win.

Calculate Probability Of Slot Machines

People sometimes think that some machines are hotter than others in the way that they are more likely to give a jackpot because more people have played them before, when in fact it is not true at all! Because the reels are independent events and each lever pull is a new event every time, your odds of winning a jackpot will remain 1:262143, even if you play for 12 hours straight non-stop.

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:

Calculate Probability Of Slot Machines

• Let’s say we use a 3-reeled machine that has 21 symbols this time on each one.
• First the lever is pulled
• The computer generates three random numbers. For example: 12,34,56. The first number tells which symbol will appear on the first reel and so on.
• If you noticed, we have 21 symbols, so why are there numbers like 34 or 56? This is because there actually 4 reels, 3 mechanical and 1 virtual, in the matrix…
• The virtual reel has 64 numbers from 1 to 64 (could also be 32 or 128 and so on). Each of these 64 numbers match to one of the 21 symbols in each reel.
• But that would mean some of the 21 symbols will have more numbers from the virtual reel than others. Exactly, that means so symbols are more likely to appear than others. So your chance of getting a jackpot number in one reel is not 1 out of 21, but even less.
• What casinos will do is have the symbols that appear before and after the jackpot symbol on each reel to be those symbols with more numbers on the virtual reel. It gives the false idea to players like us that we are so close to winning, and so we need to play more until we get a Jackpot, but in reality you’re being a tricked and are losing a lot of money.

So next time your friends abandon you at the bar while they make millions, or lose them, at the Blackjack or Poker table, you can entertain yourself with a slot machine, but will be more prepared to stop if you think you’re close to winning after 20 pulls.

(some random dude on Google :P)