how do slot machines work?

Slot machines are the most popular game on the casino floor. According to the latest report from the Nevada Gaming Commission, the state has more than 100,000 slot machines compared to fewer than 5000 gaming tables. Gaming tables can accommodate more players, but there are still more than four times as much seats for slot players as there are for table games players.

The tremendous popularity of slot machines might lead a person to assume that the games are well understood. This may not be a correct assumption. Slot machines are subject to more myths and misunderstandings than all other games on the floor combined.

These myths and misconceptions are strangely often specific to slot machines. One rarely hears that video poker is not a random game because a single pair occurs more often than a royal flush. Perhaps this is due to a deck of cards being better understood than spinning reels in a box or on a screen. Perhaps it is because slot machine manufacturers provide no information about the inner workings or even the general operation of slot machines for gamblers.

Neither slot machine manufacturers nor casinos provide even a basic explanation of how slot machines work. Slot machine manufacturers offer players very little information about their games. Their focus is game operators, like casinos. Casinos will go to great lengths to ensure that players know the rules and bet options for table games, but they offer very little for slots. The advice casinos do offer is of questionable value.

So how do slot machines work?

Understanding how slot machines work is a bit of a good news/bad news situation. The good news is that slot machines are simple devices. A player initiates the reel spin by pressing the play button or pulling the handle. The reels then stop and the game evaluates the game result to determine whether or not the player won a prize. Slot machines work exactly as they appear on the outside.

This explanation is insufficient though. It does not explain why slot machines cannot be hot or cold. It does not explain why a slot machine cannot be due. It does not explain how random chance leads to a long term profit for the casino. The question is thus not "how does a slot machine work?", but "why does a slot machine work?" This is where the aforementioned bad news comes in: it is impossible to understand why a slot machine works without understanding the math behind the game. Not to worry though, slot machine math is surprisingly simple.

gears, levers, and springs

The earliest slot machines were entirely mechanical. Some modern games are referred to as “mechanical,” but these games use stepper motors and computers to control the reels and handle all game functions. True mechanical slots used rods, gears, and springs to not only spin the reels, but also pays wins and jackpots. The mechanisms themselves are very complicated, but concept is easy to understand. The machine does all the work. The player's only responsibilities are to insert money, pull the handle, and collect any winnings.

As mechanical games use physical stops—the discrete positions that the reel can stop at—on the reels, it is very easy to see why these slot machines work and how they can produce a reliable profit for the operator despite being based on random chance.

Analysis of a mechanical game is straightforward:

1. determine the probability of each winning combination
2. multiply the probability of each winning combination by its listed pay from the paytable to find its return
3. add up all the returns for each combination to find the theoretical return to player

probability

The total number of outcomes will be equal to the number of combinations possible for the machine. With a mechanical game, the number of combinations will be determined solely by number of stops on the reels. This will be equal to the total number of symbols and blanks on the reel.

Each reel has 20 symbols and no blank spaces, so the total number of stops is 20. The total number of combinations possible would be:

The total number of outcomes is 8000. The number of ways for each winning combination will be found the same way.

A small problem emerges: slot machines only pay the highest possible value. For example, any two cherries on the first two reels will pay 3 credits unless the third reel is a lemon or bell. The number of actual combinations for cherry-cherry-any cannot include cherry-cherry-lemon and cherry-cherry-bell pays. To account for duplicated wins, the number of duplicated combinations is subtracted from the total combinations.

The above table debunks one of the slot machine misconceptions above: "slot machines cannot be random because low paying symbols appear more often than high paying symbols." Slot machines are designed so that low paying wins happen more often while high paying wins are rare. This does not mean that slot machines are not random. Slot machines are random, but outcomes are weighted for more lower pays. Cherry wins account for 980 of the 1180 ways to win on the Mills slot machine. 83% of wins will be 5 credits or less. Nearly half of all wins—49.8%—will be 3 credit cherry-cherry-any wins.

With this in mind:

The probability of landing bar-bar-bar on the payline is 0.00025. How many games would someone expect to play—the plays per hit—before landing a bar-bar-bar win?

The expected number of plays per hit can be found by taking the inverse of the probability or by dividing total number of outcomes by the actual number of winning combinations.

The jackpot bar-bar-bar combination is expected one time every 4000 games. That is not very good for a lousy 20 credits.

“But I played 4000 games and never won the jackpot!” Plays per hit is only the average value based on the probability of landing the combination. Slot machines select symbols at random. Anything can happen in the short term. A player can win the jackpot multiple times in a row on a slot machine while another can go 10,000 games without winning the top prize. However, in the long term, the jackpot is expected to be awarded one time per 4000 plays.

The plays per hit for each winning combination can be calculated in the game way. How many plays are expected for each cherry win?

The hit rate for game is equal to the actual number of winning combinations divided by the number of outcomes. This machine has 1180 winning combination out of 8000 total combinations.

The player is expected to win 14.75% games played, which means that the player will win about every 6.7797 games played.

return

The return is the amount that a winning combination is expected to pay the player. The return for each winning combination can be found by multiplying the probability of it occurring times its pay amount. The return is usually written as a percentage of credits wagered, so the amount would be divided by the total number of credits played. The Mills slot machine is limited to one credit wagers, so the return is always divided by one.

The probability of bell-bell-bell is 0.001125. bell-bell-bell pays 18 credits.

The same calculation can be done for each of the remaining winning combinations.

Calculating the returns for the cherry wins.

The bar-bar-bar win pays 20 credits plus the accumulated jackpot, but, as there is no way of knowing the amount of the average jackpot, it is not possible to calculate the specific return for jackpot win. Unlike modern progressives that use a meter, the jackpot is reliant on a coin hopper and backup hopper to refill it after wins. It is possible that the hoppers are not filled. Ignoring the jackpot for the time being, the return of bar-bar-bar is:

The return of bar-bar-bar is 0.005 without the jackpot. This is the minimum value. Any coins in the jackpot hopper would increase this. However, the theoretical return would never fall below 0.005.

The sum of all the individual returns is equal to the theoretical return to player. When someone says that slot machine machine payback is set to some number—usually as a percentage, such as 94%—this is the value that person is referring to. As with plays per hit, this theoretical return to player represents the average payback percentage in the long term. Each spin will pay an average of 78.5% of the wager amount over an infinite number of games. The actual amount returned to the player may never equal 78.5%, but the return to player will trend toward this value as more and more games are played.

The theoretical return can also be calculated using coin in and coin out. The theoretical return to play is equal to the total coin out divided total coin in for one cycle. Coin out for each combination is equal to the actual number of combinations multiplied by the prize amount.

bell-bell-bell has 9 actual combinations and pays 18 credits.

The total coin out for bell-bell-bell is 162.

There are 8000 combinations at 1 credit per play, so the coin in is 1 * 8000. The total coin out is 6280.

This is the same theoretical return found by multiplying the pay amounts by their probabilities. That is a good sign that the math is correct.

The theoretical RTP is determined by the probability of wins and their pays. The theoretical RTP can be increased or decreased by changing either the probability of winning, the pay amounts, or both. It cannot be changed willy-nilly though. The casino cannot set the game to any value it wants. Slot manufacturers can only offer choices based on probability and the paytable.

Is it really this simple?

This all seems too easy. Slot machines cannot be this simple, right? Slot machines are this simple. This is why slot machines are such reliable moneymakers for the casino. Over the long term, the odds are in favor of the house. Players may win large amounts in the short term, but the house wins in the end.

The script below runs the game though a single cycle. Each one of the 8000 different combinations will be returned one time. The results are exactly the same as the calculated return to player.

Still not convinced? Maybe you think that this would not work in the long term or that the slot machine needs to adjust game results to achieve the set RTP. The script also has a feature to play 1 million games using random numbers to determine the reel positions. The player can win the short term, but no one can beat probability in the long term.

Slot machines really are this simple. This is why they are such reliable money makers for the casino. The games may cost the casino money in the short term, but no player can beat probability in the long run. The game will trend towards it theoretical return to player over time. The Mills game will be very close to a 78.5% return to player in the long term.

That may be how slot machines worked in the past, you mutter yourself. It is not how they work today though! Modern games are programmed to have hot and cold streaks. They are programmed to have a “take mode” to recover after big wins. This is somewhat true, but not how people think. Modern games are much more complicated, but the general operation is still the same. Modern games are still using random selection to determine reel positions. The return is still based on probability and payout. There some important changes that need to be considered though.

The modern slot machine

The concept of the modern slot machine is very similar to the original mechanical machines. The clockwork mechanism has been replaced by stepper motors or a video screen while the control of the game and calculation of payout is handled by a computer processor. The overall function is still very similar to mechanical slot machines though.

Modern slot machines come with a report commonly referred to as a “PAR sheet.” PAR stands for probability and accounting report. This sheet includes the same math done above for the Mills game. These PAR sheets are considered proprietary information and not made available to the general public. Luckily, some have leaked out and can be found the internet.

The PAR sheet for Bally's Blazing 7s Double Jackpot is one of those that is available online. Blazing 7s Double Jackpot is a three reel stepper game with 22 physical stops. If the game has 22 stops on three reels, then the number of combinations is as simple as 22*22*22 = 10,648, right?

How can a game with 22 stepper stops have 72 stops on each reel? What is going on here?

virtual stops

Modern slot machines have done away with physical mechanical controls. Modern “mechanical” games are actually controlled by computers. They represent the end of the evolution from mechanical to electromechanical to digital. The use of computers and stepper motors to control reel spins has allowed slot makers to go beyond the limits of the physical reels. Virtual stops exist only in the game's programming. These internal reels can be any length. The results will be mapped to the game's physical reels. It is impossible to determine the odds of a modern slot machine from the visible reels alone.

probability

Probability for a winning combination is still calculated the same way as it was on the mechanical game. The difference is that the numbers on the reels will not be use in those calculations.

The first step is to determine the probability of each winning combination based on the paytable

Probability of each winning combination and the number of total combinations are still calculated the same way.

The probability of the other winning combinations is calculated in the same way.

plays per hit

The top line prize on the Blazing 7s Double Jackpot machine is expected to land one time per 93312 games played. The Blazing 7s Double Jackpot top prize is expected to hit much less frequently than the Mills bar-bar-bar top line pay, but the B7-B7-DJ combination pays 60 times more than the Mills bar-bar-bar pay (ignoring the jackpot).

return

Bally Blazing 7s Double Jackpot, like most modern games, allows the player to choose the number of coins wagered in the game. The paytable shows that each credit wagered has a different payout, so the return needs to be calculated for each level separately.

The formula to calculate the return has been updated slightly so that the return is a percentage of credits wagered instead of credits returned. This makes it easier to compare returns across different bet levels and games.

The return increases slightly with the number of coins bet. This is not always the case, but is generally true on modern slot machines.

This specific version of Bally Blazing 7s Double Jackpot has a theoretical return to player of 91.9640561% for one credit wagers, 92.5194465% for two credit wagers, and 92.7045766% for three credit wagers. Betting three credits per game will return slightly more money back to the player in the long term than one or two credit wagers.

Once again, coin in and coin out can be used to calculate the theoretical return to player.

choosing stops

Digital slot machines do not rely on the physical reels for anything other than showing the selected combination of symbols. This means that the machines must use something other than physical stops to determine outcome. This is where the misunderstood random number generator (RNG) comes in.

The random number generator, which can be hardware or software, produces randomly generated numbers that are then manipulated so they can be mapped to the game's reels. This is normally done through taking the remainder—the modulo—that results when the random number is divided by the number of virtual stops. For Blazing 7s Double Jackpot, the divisor (bottom number) would be 72 because the game has 72 virtual stops.

The random number generator is constantly running in a modern slot machine. The RNG cycles thousands of times per second. The exact game result that occurs is specific to that instant. A change in timing of even a microsecond is enough to produce a different result.

verifying blazing 7s double jackpot

One cycle of Blazing 7s Double Jackpot should result in the values calculated above. Does it?