• Appendices
• Slots Analysis
• Miscellaneous

1. Probability Of Slot Machines
2. Calculate Probability Of Slot Machines
3. Probability Of A Winning Slot Machine
4. Probability Of Slot Machine Formula

Slot machines are just about the only game in the casino where the odds are not quantifiable. In other words, the player doesn't know how the game was designed, so it is difficult to look at an actual game to use as an example. The probability of you getting the jackpot is exactly the same, however, whoever is watching that machine could easily make it so that you only got it one in every one-hundred-thousand times. This computer chip method is used in modern slot machines today. Basically, when it comes to slot machines, strategy boils down to this: Know the rules, your probability of winning, and the expected payouts; dispel any myths; and quit while you’re ahead. If you win $100, cash out $50 and play with the rest, for example.

Introduction

When it comes to gambling, the easier a game is to understand the worse the odds usually are. This is certainly the case with slot machines. Playing them is as easy as pressing a button. However, between the high house edge and fast rate of play, there is no quicker way to lose your money in a casino.

Before going further, let me make clear that this page addresses the way slot machines work in most parts of the United States and the world. However, some parts of this page do not apply everywhere. For example, I state that slot machines have a memory-less property, where the odds of every spin are the same. In some places, like the UK, some machines in bars, called 'fruit machines,' have a mechanism that guarantees a certain profit over the short run, which causes the game to go through loose and tight cycles. These games do not have the usual independence property of the major slot makers.

How They Work

Whether you're playing a 3-reel single-line game or a 5-reel 25-line game, the outcome of every bet is ultimately determined by random numbers. The game will choose one random number for each reel, map that number onto a position on the reel, stop the reel in the appointed place, and score whatever the outcome is. In other words, the outcome is predestined the moment you press the button; the rest is just for show. There are no hot and cold cycles; your odds are the same for every spin on a given machine.

Slot machines are just about the only game in the casino where the odds are not quantifiable. In other words, the player doesn't know how the game was designed, so it is difficult to look at an actual game to use as an example. So, to help explain how they work, I created the Atkins Diet slot machine (link). It is a simple, five-reel game with a free spin bonus round, much like IGT's Cleopatra game.

For information on how it works and all the odds, please visit my Atkins Diet par sheet.

For a more complicated example, featuring sticky wilds in the bonus, please try my Vamos a Las Vegas slot machine.

For information on how it works and all the odds, please visit my Vamos a Las Vegas par sheet (PDF).

Odds

The following table shows the casino win for Clark County Nevada (where Las Vegas is) for all slots for calendar year 2012. They define 'slot' as any electronic game, including video poker and video keno. I've found video keno to be about equally as tight as reeled slots, but video poker has a much higher return. So, the return for reeled slots should be higher than these figures.

Clark County Slot Win 2012

Source: Nevada Gaming Control Board, Gaming Revenue Report for December 2012 (PDF, see page 6).

Most players play penny video slots. Based on past research, I find the house edge on those to usually be set from 6% to 15%. In general, the nicer the casino, the tighter the slots.

Advice

While there is no skill to playing slots, there is some skill in selecting which machine to play and ways you can maximize your return. What follows is my advice, if you must play slots at all.

• Always use a player card. Slots may be a lousy bet, but the casinos treat slot players very well. A $1 slot player will probably get comped better than a $100 blackjack player. Of course, don't play for the reason of getting comps. You'll give them a lot more than they'll give you.
• The simpler the game, the better the odds. The fancy games with big signs and video screens tend to not pay as well as the simple games. However, slot players always tell me the fancy games are more fun.
• The higher the denomination, the better the odds. For that reason, it is better to play one coin per line on a 5-cent game than five coins per line on a 1-cent game.
• Don't forget to cash out and take your ticket when you leave. It is easy to forget after hitting a jackpot.
• Try to play slowly and as little as possible to get your fix.
• In some games there is a skill feature, like Top Dollar. In such games, advice is usually offered, which you should take.

Myths and Facts

Just about everything that players believe about slots is untrue. Here are the most common myths and facts. As a reminder, this page is based on slot machines commonly found in the United States. Some machines, like 'fruit machines' found in the United Kingdom work differently.

• Myth: Slot machines are programmed to go through a cycle of payoffs. Although the cycle can span thousands of spins, once it reaches the end the outcomes will repeat themselves in exactly the same order as the last cycle.

  Fact: This is not true at all. Every spin is random and independent of all past spins.

• Myth: Slot machines are programmed to pay off a particular percentage of money bet. Thus, after a jackpot is hit the machine will tighten up to get back in balance. On the other hand, when a jackpot has not been hit for a long time it is overdue and more likely to hit.

  Fact: As just mentioned, each spin is independent of all past spins. That means that for a given machine game, the odds are always the same. It makes no difference when the last jackpot was hit or how much the game paid out in the last hour, day, week, or any period of time.

• Myth: Machines pay more if a player card is not used.

  Fact: The mechanism that determines the outcome of each play does not consider whether a card is used or not. The odds are the same with or without one.

• Myth: Using a player card enables the casino to report my winnings to the IRS.

  Fact: That makes no difference. If you win $1,200 or more they will report it either way. If you have a net losing year, which you probably will, at least the casino will have evidence of it. Such annual win/loss statements may be used as evidence to declare offsetting loses to jackpot wins.

• Myth: The slot department can tighten my game with the press of a button remotely. Thus, you better be nice to the staff and tip them well, or they will use a remote control to have the machine take you down in a hurry.

  Fact: There is now some truth to the myth that the odds of a machine can be changed remotely. Such 'server-based slots' are still experimental and in a minority. Even with server-based slots, there are regulations in place to protect the player from the perceived abuses that could accompany them. For example, in Nevada a machine can not be altered remotely unless it has been idle for at least four minutes. Even then, the game will display a notice that it is being serviced during such changes. (source) Meanwhile, for the vast majority of slots, somebody would physically need to open the machine and change a computer chip, known as an EPROM chip, to make any changes.

• Myth: The machines by the doors and heavy traffic flow areas tend to be loose while those hidden in quiet corners tend to be tight.

  Fact: I've studied the relationship between slot placement and return and found no correlation. Every slot director I've asked about this laughs it off as just another player myth.

• Myth: Slots tend to be looser during slow hours on slow days of the week. However, when the casino is busy they tighten them up.

  Fact: Nobody would take the trouble to do this, even if he could. The fact of the matter is the casinos are trying to find a good balance between winning some money while letting the player leave happy. That is best achieved by slots loose enough to give the player a sufficiently long 'time on device,' as they call it in the industry, with a reasonable chance of winning so he will return to the same casino next time. If the slots are too tight, the players will sense it and be unlikely to return.

  The kind of place you're likely to find tight slots are those with a captive audience, like the Las Vegas airport. So, if the slot manager feels that 92% is the right return for a penny game, for example, he is likely to set every penny game all that way, and keep them that way for years.

Play

Probability Of Slot Machines

Reviews

• Dazzle Me (NetEnt)
• Mr. Vegas (Betsoft)
• Sparks (NetEnt)

Internal Links

• Appendix 1 shows the details and analysis of almost 4000 actual spins on a Reno slot machine.
• Appendix 2 shows an example of the virtual reels behind a hypothetical slot machine and how the average return is calculated.
• Appendix 3A: 2003 Las Vegas slot machine rankings.
• Appendix 3B: 2002 Jean/Primm slot machine rankings.
• Appendix 3C: 2002 Tunica slot machine rankings.
• Appendix 3D: 2002 Henderson/Lake Mead slot machine rankings.
• Appendix 3E: 2002 Quarter and dollar returns for Las Vegas slots
• Appendix 4 shows how the return is calculated for my Wizard's Fruit Slot Machine.
• Appendix 5 analysis of the 21 Bell Slot Machine.
• Appendix 6 Analysis of Red, White, & Blue Slot Machine.
• Lock and Roll analysis of the skill-based slot machine found in North Carolina.
• Deconstructing Jackpot Party analysis of the video slot machine.
• Deconstructing Lion's Share analysis of the classic MGM progressive game.
• Deconstructing Cleopatra analysis of the popular IGT game.
• Deconstructing Lionfish analysis of the slot game found on many Game Maker machines.
• Deconstructing Megabucks.
• Deconstructing the Atkins Diet slot machine.
• Deconstructing Lucky Larry's Lobstermania.
• Deconstructing Hexbreaker.
• Deconstructing Blazing Sevens.
• Deconstructing Hot Roll.
• Mystery progressives on Ainsworth slots.
• Mystery progressives on WMS slots.
• Baltimore Sun article, in which I am quoted.
• 100% Rebate on Slot Losses Promotions: When to quit playing when all losses are refunded.

External Links

• For a simplified explanation of slots, please see my companion site Wizard of Vegas
• German translation of this page is available at richtigspielen.com
• Another decent overview of how slots work and some practical advice for playing them is How Slot Machines Work at VegasClick.com.
• PAR Sheets, probabilities, and slot machine play: Implications for problem and non-problem gambling by Kevin A. Harrigan and Mike Dixon, University of Waterloo, Waterloo, Ontario, Canada. This is an outstanding academic paper that details how some popular slot machines were designed.
• PAR Sheets, probabilities, and slot machine play: Implications for problem and non-problem gambling - Academic paper based on the par sheets for some modern slot machines

Slots machines, as games of chance (rather than skill), are generally more about having fun than they are about making money.

However, there are things you can do to maximize your wins and minimize your losses. For example, by calculating a slot machine’s payout percentage, you can obtain a larger picture idea of how much money you stand to win back. Other tactics include using effective bankroll management techniques, joining a slots club to benefit from its rewards programs, and more.

What Are the Odds of Winning on a Slot Machine?

Slot machine odds used to be easy to calculate. When you’re dealing with three reels, ten symbols on each reel, and a limited pay table, then it’s just a simple math problem. But the rise of electromechanical slot machines and (later) video slots added some complexity to the situation.

How Probability Works

Probability has two meanings. One is the likelihood of whether or not something will happen. The other is the branch of mathematics that calculates that likelihood. To understand the odds as they relate to slot machines (or any other gambling game), you have to understand the basic math behind probability.

Don’t worry though. The math isn’t hard. Probability involves addition, subtraction, multiplication, and division, all of which you learned in middle school.

The first principle of probability is that every event has a probability of between 0 and 1. If something has no chance of ever happening, then its probability is 0. If something will always happen, no matter what, then its probability is 1.

Probability is, therefore, always a fraction. It can be expressed in multiple ways, as a decimal, as a fraction, as a percentage, and as odds.

A simple example is a coin flip. The probability of getting heads when you flip a coin is 50%. That’s common sense, but how is it determined mathematically?

You simply take the total number of possible outcomes, and divide the outcome you’re trying to determine the probability of it by that number. There are two possibilities when flipping a coin, heads or tails, but only one of them is heads. That’s 1 divided by 2, which can be expressed as ½, 50%, 0.5, or 1 to 1 odds.

Odds are expressed as the number of ways something won’t happen versus the number of ways that something will happen. For example, if you’re rolling a single six-sided die, and you want to know the odds of rolling a six, you’re looking at 5 to 1 odds. There are five ways to roll something other than a six, and only one way of rolling a six.

When you want to determine the probability of multiple things happening, you use addition or multiplication, depending on whether you want to determine whether one OR the other event will occur, or whether you want to determine whether one event AND the other event will occur.

If you’re looking at an “OR” question, you add the probabilities together. If you’re looking at an “AND” question, you multiply the probabilities by each other.

So if you want to know what the probability of rolling two dice and having one or the other come up with a six, you add the probabilities together. 1/6 + 1/6 = 2/6, which is rounded down to 1/3.

If you want to know the probability of rolling two dice and having BOTH of them come up six, you multiply the probabilities. 1/6 X 1/6 = 1/36.

How Slot Machine Odds USED to Work

Early slot machines were mechanical devices. They had three metal reels that had ten possible stops each.

To calculate the odds of a single symbol appearing on a reel, you just divide the one symbol by the total number of potential outcomes. So if you had one cherry on a reel, your odds of hitting that cherry were 1/10, or 10%.

To calculate the odds of getting three cherries, you multiple 1/10 X 1/10 X 1/10 and get 1/1000, or 0.1%.

If the odds of hitting that symbol are the same as all the others, then you have 10 possible jackpots you can win, which means that your chances of winning SOMETHING are 10/1000, which is 1%.

Most people wouldn’t play a slot machine that lost 99 times out of 100, though, so slot machine designers added additional, smaller prizes for getting two symbols out of three for certain symbols. And as long as they paid out less in prizes than the odds of hitting those jackpots, then those slots are guaranteed to make a profit in the long run.

For example, if a prize for hitting three cherries was $1000, you’d be playing a break-even game, but if the prize were $750, it’s easy to see how the casino would be guaranteed a profit. The difference between the odds of winning and the payout odds is where the casino makes its money.

How Slot Machines Work Now

Modern slot machines use a computer program called a random number generator to determine the outcomes of the various spins of the reels. This creates an imaginary reel with a number of symbols limited only by the program in question.

A mechanical slot machine with 256 symbols per reel would be huge, too large to play, much less to build. But a computer can create an imaginary reel with 256 symbols per reel and take up no more space than an iPod Shuffle.

To make things even more interesting and entertaining, slot machine designers can program different probabilities for each symbol to come up. Most symbols might come up once every 256 spins, but others might come up twice as often, while still others might only come up half as often.

This enables slot machine designers and casinos to offer slot machine games with far larger jackpots than they were able to when they were limited by mechanical reels. And they’re able to offer these large jackpots and still generate a healthy profit.

How Does This Relate to Payback Percentages?

The payback percentage is the amount of money that the slot machine is designed to pay out over an enormous number of spins. This number is almost always less than 100%. The difference between 100% and the payback percentage is the house edge, and that’s where the casino makes its profits.

A simple example can help illustrate how this works. Suppose you have a slot machine with three reels with ten symbols on each, and it only pays out when three cherries hit. The odds of winning that jackpot, as we determined earlier, is 1/1000.

If we set the jackpot as $900, and charge $1 per bet, the payout percentage for that game will be 90%, or $900/$1000. Of course, no one would play a slots game which only paid out once in every 1000 spins, which is why there are various smaller payouts programmed in.

There’s no way to tell what the payback percentage on a particular game is unless you have access to the par sheet for that machine. Casino management has that information, but players never have access to that info.

Calculate Probability Of Slot Machines

The best slot machine odds are almost always found in real casinos. If you see slot machines in an airport or a bar, be aware that the payback percentages on those games is much lower than you’ll see in a real casino.

How to Win at Slot Machines

Everyone would like to know how to win at slots, but the truth is that winning at slot machines isn’t any harder than losing at slot machines. You put your money in the machine, spin the reels, and hope for the best. Slot machines are meant to be fun; they’re not intended to provide the player with an income.

In fact, the reality is just the opposite. Slots are there to provide the casino owners with an income. How that works is one of the subjects of this page.

On the other hand, you can minimize your losses and increase your enjoyment of slots games by understanding how they work. You can also learn which slots pay back the most money. In the long run, the house will still have an edge over you, but understanding how much you can expect to lose in a given venue can help you make better bankroll management decisions.

In fact, it might be a good idea to modify you definition of “winning at slots”. Instead of considering yourself a winner if you bring home a big profit, consider yourself a winner any time you played and had a lot of fun.

How Slots Work

All slot machines in modern casinos use a random number generator (an “RNG) to determine the results of each spin. An RNG is a tiny computer that does nothing but constantly generate numbers. When you push the spin button, that microcomputer selects a number which determines the outcome. In fact, this happens before the reels have even stopped spinning.

On modern slot machines, the reels are just there for show. From a practical standpoint, you could put a quarter in a machine, push a button, and have the screen flash: “You lose!” or “You win $10”. The mechanism that determined the outcome would be the same, but who would want to play a game like that, especially if you know that the house has a mathematical edge over the player.

The spinning reels, the sound effects, and the bonus games are all there to make the game more interesting to play. If you don’t like the artwork, the music, or any other aspect of a slots game, don’t bother playing it, because those are the real rewards of playing. The chance of getting lucky and winning a jackpot is a real reward, too, but don’t ignore the other aspects of the game.

The random number generator is programmed to pay back a certain percentage of the money paid into it over a period of time. This period of time is known in gambling math as “the long run”, and it’s a lot longer than most people think. We’re talking about tens of thousands of spins, not dozens or hundreds.

This percentage that’s programmed into these machines is always less than 100%. If a slots game were programmed to pay back more than 100% of the money put into it, it would lose money for the casino.

Casinos aren’t in business to lose money.

The trick is to find slot machines that have the highest payout percentages.

Which Slots Pay Back the Most Money

If every slot machine game in the world had a payback percentage posted on the machine somewhere, it would be easy to determine which slots pay back the most money. You could limit your play to machines with a payback percentage of over 95% for example.

It’s too bad casinos don’t provide that information on specific games, though.

Probability Of A Winning Slot Machine

You can find information about specific locations and their payback percentages, though. Some gambling guides and magazines publish this information. For example, The American Casino Guide provides certified information about the payout percentages in various states. Not all states reveal this information, but it’s not a huge leap of logic to expect better payback percentages in states that do reveal this information.

For example, the overall payback percentage for slots in Black Hawk, Colorado is 92.8%. In Central City, Colorado, it’s 92.93%, and in Cripple Creek, it’s 93.66%. Alabama doesn’t release the numbers on their payback percentages.

Which casinos do you think offer the better game?

A couple of guidelines hold true no matter where you play, though. One of those is that payouts are better in large cities with lots of gambling. For example, the payouts in Vegas are higher overall than the payouts in Colorado. And the payouts improve when you play for higher stakes. For example, penny slots in Vegas average around 88% to 91%, but dollars slots average between 93% and 96%. Finally, slot machines at airports usually offer the lowest payouts.

What does that mean for the player? It means that over the long run, if you wager $x on a particular game, you’ll win back $x times the payback percentage for that machine. If you’re playing a dollar slot machine on the Strip in Las Vegas, for example, and the payout percentage is around 93%, then if you place $10,000 in wagers, you’ll win back $9300. You lost $700.

Probability Of Slot Machine Formula

That’s only a long term mathematical expectation, though. In the short run, anything can happen, and that’s what keeps people playing.

How to Maximize Your Winnings and Minimize Your Losses

There are three ways to maximize your winnings and minimize your losses. The first is to always join the slots club, and always use your member card while you play. Slots club members get a percentage of their play returned to them in the form of casino rewards and cash back. This is normally a tiny percentage (think 0.1% or 0.2%), but it adds up, especially if you play a lot.

Don’t buy into the myth that playing with your slots club card lowers your expected return on the game, either. That’s not true. The random number generator in these games has no way of knowing whether or not you’re using your slots club card or not.

The second way to increase your winnings and minimize your losses is to use effective bankroll management techniques. This means limiting the amount of time that you play, limiting the amount of money that you’re willing to lose in any session and in any given gambling trip, and finding other fun things to do with your time besides just playing the slots.

Finally, try to play the machines with the highest payout percentage. Over the long run, if you keep playing, you’ll probably eventually wind up a loser at the slots (unless you hit a huge progressive jackpot), but you’ll lose your money more slowly and get more entertainment value for the money you gambled.