Slot Machine Probability: a Mathematical Approach to Slot Machines
Slot machines are Casino games with no real winning strategy.
Here on CasinoSmash we explore the intricacies of winning at Slot machines in a guide on how to win at Slots where we highlicght how everything depends on your luck.
On that guide, you discover how to use each game's payback percentage to pick what Slots to play when you gamble online, you play at mobile Casinos, or you are in Las Vegas for some good, old 'analogic' fun.
Today, however, you are about to learn something different.
You are going to understand the immense importance of the Random Number Generator and how you can use Slot machine math to understand chances of winning at Slot machines.
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Table Games vs Slot Machine Odds
Although the house edge is present also when you play Roulette, or you play Blackjack, the nature of Casino table games is different.
Each time you enter a game, you are requested to make decisions that can somehow influence your odds of hitting a win.
The Casino wins more often if you place single number bets when you play Roulette and if you decide to 'Hit' on a 17 when you play Blackjack. Likewise, you win are more likely to succeed if you play Red / Black on Roulette and if you 'Stand' on 17 points.
That's obvious. But that's not how Slots work.
You can't apply the same mathematical approach to beating Slot games as none of the modern Slot machine games work like that.
Video Slots are a negative equity game or a –EV game, as professional gamblers prefer to say.
Slots are -EV: What does it mean for you?
The probability to win at Slots in the long term is not high . It depends on how frequently a machine pays, what its RTP is, and if you can stop as soon as you win. If you don't, the Casino wins. Always.
When you play online Slots at Casinos that are certified by international authorities and get audited by third-party independent companies, you can hit the Slot machines safe in the safe in the knowledge that the odds of hitting winning combinations are in the hand of a certified and fair Random Number Generator (RNG).
The Random Number Generator
Each time the lever is pulled, a button is pushed, and the reels are set in motion, a random combination of symbols is selected by the game. That's how Slots work.
The random selection feature ensures that every spin is independent. Thus the results of the previous spins do not affect the next ones in any way.
The probability of winning when you hit the Slot machines is determined by the number of reels and the number of symbols on each reel. And whether a randomly generated pattern will get you a winning combination of symbols.
While it may seem like an impossible task, some tricks can make your spins work in your favour. Below, you will find the ultimate guide on how to use probability to win at Slots.
The Probability of Winning on a Slot Machine
The number of possible combinations on any Slot machine is easy to calculate. You need to multiply the total number of symbols (stops) each Slot (reel) has.
Slot Machine Math: Basic Calculations
On a three-reel machine with six symbols apiece, the total number of possible combinations is 6 x 6 x 6 = 216 combinations. Similarly, in a three-reel machine with 20 symbols each Slot, the total number of combinations would be 20 x 20 x 20 = 8,000.
Calculating the odds of winning on a Slot machine is also simple. You need to divide the number of winning combinations by the number of possible combinations. That is:
How to Calculate Slot Machine Odds: :
Number of winning combinations / Number of possible combinations = your machine odds
The highest wins come with challenging combinations to hit — especially when you play to hunt down the progressive jackpots.
A Theoretical Approach to Slot Machines
(To write this guide to Slot machine probability, we will assume that every Slot machine has a single line and only accepts one coin per roll. Obviously, it's impossible.)
The first machine has three reels. On each reel, there is a banana, a lemon, an orange, an apple, a melon, and a joker. The payouts are as follows:
- OPTION 1: Three jokers - 30 Coins
- OPTION 2: Any 2 of a Kind Fruit - 10 Coins
- OPTION 3: Any Pair of Jokers - 4 Coins
- OPTION 4: Any Single Joker - 1 Coin
The number of possible combinations is: 6 × 6 × 6 = 216
So, How Easy Is It to Win at Slots?
Depending on which of the previously mentioned four outcomes you're aiming for, there are different probabilities for collecting some cash, which we will explain below.
Option 1: 30 Coins aka. The Big Win
You have only one chance to bring home the machine’s most significant win.
Option 2: 10 Coins
Three-of-a-kind that doesn't include a joker. On the first reel, you can get any of the five fruits (banana, lemon, orange, apple, or melon).
Once the fruit is chosen, it has to also appear on the paylines in the next two reels. If you do the math, you can see that there can only be five such combinations: three bananas, three lemons, three oranges, three apples, and three melons.
Option 3: 4 Coins
The jokers can appear on any two reels: reels 1 and 2, reels 1 and 3, and reel 2 and 3. To get this winning combination, the reel that doesn't feature a joker symbol must contain a fruit. Hence, the total possible winning combinations are 1 × 1 × 5 + 1 × 5 × 1 + 5 × 1 × 1 = 15.
Option 4: 1 Coin
The jokers can appear on any of the reels of the Slot machine. What is important, however, is that the other two reels have a fruit on them. This winning combination is more common as the right calculations suggest 75 different scenarios can lead to it (1 × 5 × 5 + 5 × 1 × 5 + 5 × 5 × 1 = 75).
All in all, the Slot machine we used for this quick example features 96 different winning combinations: (1 + 5 + 15 + 75 = 96).
Let us calculate the total Slot’s payout using the table below.
|Winning Combinations||No. of Comb||Win||Payoff for One Coin||Percentage payoff|
|Any Three Similar Fruits||5||10||50||23.27|
|Any Two Jokers||15||4||60||27.91|
|Any One Joker||75||1||75||34.89|
Payout percentage = (1 × 30 + 5 × 50 + 15 × 4 + 75 × 1) ÷ 216 —> 215/216 = 0.9954 or 99.54 percent
A payoff of 99.54 percent is quite good for a Slot game. However, it can be a little misleading given that harder combinations are given more weight.
If everything were paid equally, about 44.44 percent or 96/216 of combinations would win.
Analyzing a Complex Slot Machine
Note: Also in this example, we will use one line, three reels
The distribution of symbols on each reel is as follows:
|Symbols||Reel 1||Reel 2||Reel 3|
Note that there is a non-equitable distribution of symbols on each reel. With the lack of Sevens (7s) Cherries and BARs, it is clear that the probability of getting winning combinations involving these three symbols will offer high payoffs.
The payouts and winning combinations are as follows:
- 3 BARs wins 60 coins. Number of winning combinations: 1
- 3 Sevens win 40 coins. Number of winning combinations: 3 (3 × 1 × 1 = 3)
- 3 Cherries win 20 coins: Number of winning combinations: 36 (4 × 3 × 3 = 36)
- 2 Cherries Wins 4 coins: Can be anywhere on the reels
- TOTAL for Fruits: 540, For Oranges 5 × 6 × 6 = 180, Bananas 5 × 6 × 6 = 180 and Lemons 5 × 6 × 6 = 180.
Total number of winning combinations:
- Cherry, cherry, other: 4 × 3 × (23-3) = 240
- Other, cherry, cherry: (23-4) × 3 × 3 = 171
- Cherry, other, cherry: 4 × (23-3) × 3 = 240
- TOTAL COMBOS: 651
Cherry in any one reel wins 1 coin. Meaning:
- First Reel: 4 × 20 × 20 = 1600
- Second Reel: 19 × 3 × 20 = 1120
- Third Reel: 19 × 20 × 3 = 1120
- Total: 3,880 combinations.
The table below shows how the machine’s payout percentage is calculated:
|*Winning Combinations||No. of Comb||Win||Payout per Coin||Percent of Payout|
|Three of any other symbols||540||10||5,400||44.51%|
|Any Two cherries||651||3||1,953||16.10%|
|Any One cherry||3,880||1||3,880||31.96%|
The payout percentage for this Slot machine is 99.77 percent.
If everything was paid equally at one coin, it would go down to just over 42 percent. For more complex reel machines (more reels and multiple lines), the use of an Excel spreadsheet or a computer program will help in calculating the payoff as the number of calculations will grow with complexity.
The House Edge
If you are planning to play on Slot machines with the odds of winning set at 95 percent, you will be led to thinking that you can win 95 percent of the times you play at the machine.
However, the fact is that five percent of every dollar you bet will go to the house. For Slots, the house edge is higher than for any other Casino game.
How to Determine the House Edge of a Slot Machine
It is often impossible to determine the exact house edge of any given reel machine. Instead, players depend on the information that is provided publicly, such as regulated pay off ranges.
In some states, the national Gaming Boards state that most Slots must have a 73 percent return to the player. Generally, small Slot machines are closer to this threshold than the larger Slot machines. For this reason, it is advisable to skip the penny Slots and nickel Slots if your bankroll allows it.
Think of Slots like a coin flip. If a coin is flipped ten times, and for the first nine times it lands on heads, what will the tenth flip be? The chances are still 50/50 regardless of the previous results. Slots are the same in that sense.
The Best Casino to Play Slots
The last and the most important thing to know today is where to play Slot machines to get higher chances of winning.
Our recommendation, as always, is to pick internationally licensed online Casinos where the games are provably fair, and you can rely on the information available on the screen to make an informed choice.
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All the games there have precise info about essential elements like the return to player, the paylines, and the number of symbols featured in the Slot machine. To see this info, launch a game and click the Help/Rules/?/Info button.
That makes it easy to calculate the probability of getting the pay you desire and determine the Slot machine odds.
To test the machines and avoid potential losses on less paying Slots with a low payback percentage, you can use the deposit bonuses that are yours right as you open your free gaming account.
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Disclaimer: Gambling is a game of chance and there's no sure-fire formula that guarantee winnings. You can improve your winning chances by using the right in-game strategies but there is no way to predict when a machine or a Casino game is going to pay. Also, this page contains affiliate links. If you click through and play, we might earn a commission. That’s what allow us to keep CasinoSmash free and bring you the best online Casino bonuses!