Slot Machine Probability: a Mathematical Approach to Slots
- When you play modern slot machine games at online casinos, your odds of winning depends on a variety of factors. Most of them are out of the player's control.
- The way a slot pays depends on the game's Return to Player (RTP) and volatility. Slot machine payouts are always different, so your odds of winning change depending on what particular slot you play.
- This guide to slot machine probability helps you understand how to understand your odds of winning and get the best slot machine odds when you play for free or real money.
- If you are looking for a way to beat slot machines online, keep in mind that casino games are never a good option if you want to earn some cash since the odds of winning are not in the players' favor. Please, always gamble responsibly.
- Table Games vs Slot Machine Odds
- The Random Number Generator
- The Probability of Winning on Slot Machines
- A Theoretical Approach to Slot Machine Games
- Analyzing a Complex Slot Machine
- The House Edge
- The Best Casino to Play Slots
- FAQ about Slots, Mathematics, and Probability
- Other Popular Slots Articles
We all love slot machines and we all love the idea to find loose slots to crack with a good and tested strategy. Unfortunately for us, however, the way slot machines work is different - and the only way to beat modern slot machines...is to be lucky.
Despite what some self-proclaimed "online slots experts" share on their blogs or on the casino floor, playing slots is all about spinning the reels of slots games hoping to be lucky enough to hit a winning combination.
Slot machines are casino games where the entire outcome depends on luck, and no slot machine strategy can help you change that.
Here on CasinoSmash we explore the intricacies of winning at slot machines in a guide specifically dedicated to how to win at slots. If you want to know why casino sites push online slot games so much and why everything goes doesn't to good or bad luck, go read that article now.
That guide helps you discover how to use each game's payout percentage to pick what casino slot to play at online casinos or on the casino floor of your favorite brick-and-mortar venue.
This page, on the other hand, takes you on a more advanced journey that helps you understand how slot machines pay, how the Random Number Generator (RNG) runs the show, and how you can use slot machine math to understand chances of winning at slot games.
Get bonus spins and more to play high denomination slots with an RTP of 97% or more!
Table Games vs Slot Machine Odds
All casino games feature a 'house edge', an unfair advantage that helps the casino beat the players and earn money in the long term. What makes slot games differ from most other games, is that players can't manipulate or reduce the house edge like in blackjack, video poker, or baccarat.
When you gamble online on a casino game like blackjack, you make decisions that can influence your odds of hitting a win. Regardless of the payout percentage of the casino game you picked, you can get higher chances to win by applying the correct strategy or by picking the right bets.
Statistically, a casino wins more often if you play roulette placing single number bets or 'hit' on a 17 when you play blackjack. Likewise, your winning chances are higher if you play roulette betting only on Red and Black, or if you 'Stand' on 17 points when you play blackjack.
This happens because these games allow casino patrons to choose how to play, giving them the opportunity to modify the outcome of the game. Unfortunately, that's not how slots work.
If the 'correct' choices in roulette, blackjack, and other popular games to win real money online can be determined by mathematical models, the way slot machines work is just different. Online 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 is not high. It depends on how frequently a machine pays, what is the payout percentage of the games you play, and your ability to stop as soon as you win.
This, of course, assumes you only play online slots at licensed casinos where the games are certified fair by international authorities and third-party independent companies. Like we discuss on our page dedicated to our recommended online casinos, playing on legit websites is the only way to be sure all the online slots games you access are fair and not rigged.
The Random Number Generator
Each time you play slots and set the reels in motion, the machine terminates its spin with a random combination of symbols on the screen. That randomness is all you need to keep in mind to understand how online slots work.
This mechanism ensures that every spin is independent and the results of the previous spins do not affect that of the next ones in any way.
The probability of winning when you play slot machines is determined by a combination of elements that include the number of reels, the number of symbols on each reel, and the number of active paylines.
Some slot games help you manipulate these elements by allowing you to select how many paylines you want to play and how many reels you want to activate. Keep in mind that as this helps you increase your slot machine odds, it also increases the size of your bet, making playing the games considerably more expensive for casino patrons.
Not all hope is lost, though. While beating slot machines may seem an impossible task, there are some tricks that can help you get the most out of a game's payout percentage and improve your odds of winning. If you want to discover how that happens, it's time we discuss how to use probability to win at slots.
The Probability of Winning on Slot Machines
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 for 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 relatively 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 slot game odds
This seemingly unimportant formula is crucial if you intend to approach a slot game with high hopes. All too often I see players in Las Vegas or at an online casino dreaming of mega wins at progressive jackpot slots that give them close to no chance to make it.
While playing for a progressive jackpot can be exciting (who doesn't dream to become a millionaire playing slots?), maths and probability show it's close to impossible. Personally, I still think there are great penny slot machine games that are just as fun as most progressive games while offering much better odds to win.
A Theoretical Approach to Slot Machine Games
Note: in this part of our guide to slot odds and probability, we will assume that every slot machine has a single line and only accepts one coin per game. This is a clear oversimplification of the way slots on casino sites work, but it helps us to understand better how to calculate slots odds.
To better understand how a slots game work, let's look at a few examples. In the first one, we play a machine with three reels. Each reel features six symbols: a banana, a lemon, an orange, an apple, a melon, and a gold bar.
The paytable of this game are as follows:
- Max Win - Fixed Jackpots: Three gold bars - 30 Coins
- Win 2:: Any three-of-a-kind with fruit symbols - 10 Coins
- Win 3: Any pair of gold bars - 4 Coins
- Win 4: Any single gold bar - 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. Again, remember the game in the example is a non-existent and oversimplified slot that does not include bonus games, does not come with a bonus feature or a progressive jackpot.
Max Win: 30 Coins
You have only one chance to bring home the machine’s most significant win.
Option 2: 10 Coins
A three-of-a-kind that doesn't include a gold bar. 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 gold bars 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 gold bar 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 gold bars 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).
Calculating the Payout and Probability to Win
Let us calculate the total slot’s payout using the table below.
|Winning Combinations||No. of Comb||Win||Payoff for One Coin||Payout Percentages|
|Three Gold Bars||1||30||30||13.96|
|Any Three Similar Fruits||5||10||50||23.27|
|Any Two Gold Bars||15||4||60||27.91|
|Any One Gold Bar||75||1||75||34.89|
A payoff of 99.54 percent would make this a very loose slot, as this machine would pay a lot more often than any other game you find at any online casino site. Most real slots have payout percentages of around 96%-97%.
Analyzing a Complex Slot Machine
Note: Also in this example, we will use a game featuring one line and three reels
The game in our second example is a classic slot machine featuring four fruit symbols (cherry, lemon, banana, orange), one SEVEN, and one BAR symbol.
Like in most slots, the distribution of symbols on each reel depends on the payout percentages, with the best-paying symbols being the scarcest. The symbol distribution 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 payouts.
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.
The 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 per cent .
If everything was paid equally at one coin, it would go down to just over 42 per cent. 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 per cent, you will be led to thinking that you can win 95 per cent of the times you play at the machine. Which is not true.
That number indicates that five per cent 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, with no difference between higher denomination slots, low volatility slots, and high volatility slot machines. Instead, players depend on the information that is provided publicly, such as regulated payoff ranges.
In some states, the national Gaming Boards imposes that slots must offer a return-to-player rate higher than 73 per cent . When it comes to online gambling, however, we find most games start at 94%, with the majority of them featuring a RTP of 96% or more.
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.
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 licensed online casinos where the games are provably fair. When you pick one of the best online slot sites, you can rely on the information available on the screen to make an informed choice.
All the games there have detailed info about elements like the return to player, the paylines, and the number of symbols featured in any slot machine. To see all this information, launch a game and click the Help/Rules/?/Info button.
Once you can see what a game's RTP, layout, and paytable are, you can calculate the probability of getting the win you desire and determine correct slot machine odds.
Using Casino Bonus Offers to Play Online
To test the machines and avoid potential losses on a less paying slot game with a low payout percentage, you can use a casino bonus. Here on CasinoSmash we specialize in finding the best deposit and no deposit bonuses for slots and table games, as we believe these are a great opportunity to play games for free.
When it comes to slots, some casinos offer no wager free spins that let casino patrons cash out any eventual wins without too many complications. If you are interested in the best no deposit bonus offers to play slots, check out our page dedicated to real money casino free spins.
Alternatively, for a more generic approach, click here to find all the best casino bonuses and use math to beat all the top real money modern slot machine games online!
Also find out - does playing max bet increase odds to win at slots?
FAQ about Slots, Mathematics, and Probability
How are slot machine odds calculated?
Odds on slot sites are calculated considering a game's structure (number of reels and number of symbols per reel) and the paytable (number of winning combinations, distribution of the symbols on the reels).
What are the odds of hitting on a slot machine?
Slots odds vary from one game to the other. If you want to know how to calculate the odds of hitting a winning combination on a slot, you can refer to these three examples and apply the same logic to come up with your own slot machine calculator.
What slot machine has the best odds of winning?
Slot machines with a higher Return to Player (RTP) rate offer the best winning odds. When playing at online casinos, you may want to look for games that offer an RTP higher than 97%. If you don't know how to find them, here's a list of the slots with the highest RTP.
What slot machines are most likely to hit?
In Las Vegas and online, low volatility slots are considered to be 'loose slot machines' because they offer frequent wins. While these games pay more often than high volatility slots, they also pay much smaller prizes.
Can I download a free slot machine odds calculator?
Since the odds vary from one game to the other, you can't download a slot odds calculator. You can, however, use some formulas on Excel to calculate your probability to win at slots based on the characteristics of each game you play.
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