Why do slot machines work?

myths and urban legends

Slot machines are incredibly popular. Clark County, Nevada, which includes Las Vegas, has more than 100,000 slot machines. This may not seem like a very large number, but the entire state of Nevada has fewer than 5000 gaming tables and only 627 licensed poker tables (https://gaming.nv.gov/modules/showdocument.aspx?documentid=19987). Slot machines account for the majority of casino revenue. The gambling industry is built on them.

Despite this massive popularity and importance, slot machines are not very well understood by gamblers. Myths and misunderstandings about slot machines are widespread. Gamblers believe slot machines:

• have "hot" and "cold" streaks
• are programmed to have "take" and "win" modes
• that have not awarded a large prize recently are "due" for a win
• need to track wins and losses in order to ensure they can adjust returns to achieve their set payback percentage
• award new players "tastes" to keep them playing.
• are programmed to pay out at certain times and to withhold wins at other times.
• cannot be random as certain symbols and combinations appear more often than others
• may be set to return over 100% as a way of encouraging patrons to play similar games set at much lower returns

Game manufacturers do not appear to be very concerned with debunking these myths or even explaining the basics of slot machines to players. None of the major slot manufacturers provides any information about the workings of slot machines on its website.

Casinos are no better here. While casinos offer detailed explanations of table games, including best ways to play the games to minimize house edge, casinos offer nothing about slot machines. Some offer less than nothing by spreading urban legends.

so why do slot machines work?

Understanding why slot machines work is a bit of a good news/bad news situation: the good news is that slot machines are simple devices; the bad news is that slot machines are based on math.

Slot machines look complicated, but the basic functions can be described in a just a few steps:

1. The player insert money to the game using coins, tickets, or electronic means
2. The player initiates a game by pulling a handle or pressing a button
3. The slot machine reels begin to spin
4. The reels randomly stop at discrete positions to display a symbol on the payline(s)
5. The game evaluates the results and awards the player prizes for any winning combinations

This is how all slot machines operate. Modern digital games change the order slightly, but the same thing is happened when the gambler plays a new slot machine.

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 to win. It does not explain how random chance leads to a long term profit for the casino. It does not explain why certain symbols appear more often than others. This is why this article focuses on why slot machines work; this is why understanding the math behind slot machines is fortunately/unfortunately so important.

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 computer processors 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 for the player. 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 of the reel that can land on the payline—it is very easy to see not only why these slot machines work, but also why they can produce a reliable profit for the operator.

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 contribution to the return to player
3. add up those contributions 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. The number of symbols on each reel will be multiplied together for each winning combination. The winning combinations will be found on the pay table.

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 combinations appear more often than higher paying ones. This does not mean that slot machines are not random. Slot machines are random, but outcomes are weighted toward lower pay combinations. 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 for every 4000 games played.

“But I played 4000 games and never won the jackpot!” The jackpot can be won more or less often than expected. In the long term, the bar-bar-bar hit will be expected to occur in about 0.025%% of games played. This does not imply that someone who has played 3999 games without a jackpot will be guaranteed one on the next play. The odds of getting a jackpot on that 4000th game would still be 1/4000. For most slot machines, the odds of winning a jackpot are the same on every spin.

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. The expected plays per hit is 6.7797. For every seven games played, the player should expect to win one time.

return

The return is the amount of money wagered that comes back to the player. The return for each winning combination can be found by multiplying its probability of hitting by the amount it pays. 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 return values are 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 payback—"they have the machines set to 70%"—they are referring to the theoretical return to player. 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 actual 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; the above 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.

are slot machines really this simple?

This seems too easy. Slot machines produce billions in revenue per year. Slot machines must be much more complicated, right?

Slot machines are this simple. 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.

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

A slot machine may lose money in the short term, but the house wins in the long term. The more games played, the closer the Mills game gets to the theoretical return to play of 78.5%. Slot machines make the casino a little money for a very long time. After 1,000,000 spins, the 25¢ Mills game will make the operator about $50,000.

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 only difference will be where the numbers are coming from. For a modern game, the virtual stops are the basis of all calculations.

The first step is to determine the probability of each winning combination found on the game's paytable

Probability is 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?