Slot Machine Probability Distribution

Casinos generate numbers for their slot machines based on a normal distribution. So when Probability and Expected Profit/Loss are set up in the house’s favor, they can lose once or twice, but overall the casino will be profitable. All because the casino’s math calculations reflect the set distribution. The author does not just throw the slot mathematics to the audience and run away, but offers an ultimate practical contribution with the chapter 'How to estimate the number of stops and the symbol distribution on a reel', a surprise for both players and producers, where one can see that mathematics provides players with some statistical methods.

In Pokémon Red, Blue, and Yellow, there are playable slot machines in the Celadon Game Corner. The rules are apparently simple: The player needs to line up the same symbols on horizontal or diagonal lines (the set of available lines depends on the number of coins betted), and the player can stop the reels, one at a time, using the A button. In reality, the game may cause the reels to 'slip' after the A button press either to hinder or to help the player, and the rules for doing so are actually complicated, with many oddities that may or may not be glitches. (Together with the separate building for prize claims, this is comparable to how real life Japanese pachi-slots are played.)
Thanks to the disassembly, the actual behaviors of the slot machines in Generation I are completely understood, although at some places it may still be unclear whether they match the intended behaviors.

Overview

For each individual spin, the slot machine can be in one of three modes:Mode bad, which guarantees that the player cannot get any match in the available lines.Mode good, which allows the player to get matches of symbols other than sevens or bars.Mode super, which allows and actively helps the player to get matches of sevens and bars. Mode super is the only mode of the three that will persist between spins. It will only end when the player wins a reward of 100 coins (bars) or 300 coins (sevens), and in the latter case only with a probability of 50% (i.e. it still have a 50% chance to persist).In general, for each spin, the mode is set to mode good or mode super with probability 45/256 (~17.6%), with the probability of mode super further depending on which slot machine the player is playing on (see below). It is set to mode bad with probability 210/256 (~82.0%). There is also a chance of 1/256 (~0.4%) to enter a lucky streak which allows the player to play on mode good (but never mode super) for all the spins after this one, until the player wins 60 times.The function to set the mode, called before each spin

The lucky slot machine

Whenever the player enters the Game Corner, a random number between 0 ~ 31 is generated, and if it is 0, it has a 7/8 chance of becoming 1.More precisely, a random number between 0 ~ 255 is generated, it becomes 8 if it is less than 7, then it is divided by 8, discarding the remainder. See the function to generate the index of the lucky slot machine in the disassembly. Then, whenever the player plays on a slot machine, the number is compared with the index of that slot machine (0 ~ 35) plus one. If it matches, the chance of playing in mode super is set to 5/256 (~2.0%), and otherwise it is set to 2/256 (~0.8%).The function to set the chance of mode super
Effectively, this means that every time the player enters the game corner, there is usually a 'lucky slot machine' among the machines with index 0 ~ 30. The lucky slot machine is most likely (15/256, ~5.9%) to be the one with index 0 (the rightmost, bottommost one), and cannot be the ones with index 31 ~ 35 (the bottom five in the leftmost column). The other ones all have a probability of 1/32 (~3.1%) to be lucky, including the ones occupied by an NPC, or otherwise unplayable ('Out of order', 'Out to lunch', or 'Someone's keys'). Finally, there is a tiny chance (1/256, ~0.4%) that the lucky slot machine is the nonexistent slot machine -1.

Reel stopping

The mode of the slot machine comes into play when the player presses the A button to stop a reel, as the reel may 'slip' after the A press depending on its position. The exact logic for stopping the reel also depends on which reel the player is stopping. Internally, the first two reels are programmed to slip 4 times unless a condition is met earlier, while the third reel can either slip up to 4 times to help the player get a match, or slip any number of times to prevent a match.

First reel

In mode bad or mode good, the condition for the first reel to stop early is that the middle symbol is not a cherry. Since there are no two consecutive cherry symbols (in fact, there are no consecutive same symbols on any reel), that means the first reel will slip at most once.
In mode super, the condition for the first reel to stop early is that any of the three symbols showing is a symbol with an internal value less than that of the seven symbol. Since the seven symbol has the lowest internal value of all symbols, this condition is in fact never satisfied, and the first reel will always slip by 4 positions.The condition for the first reel to stop early

Second reel

In mode bad or mode good, the condition for the second reel to stop early is that there must be a 'potential match' on the first two reels, i.e. two of the same symbols that may form a line if the third reel stops in a good position. The player's bet is not considered; for example, a diagonal potential match counts even if the player only betted 1 or 2 coins.
In mode super, the condition for the second reel to stop early is that the potential match found is of either sevens or bars, or that there is no potential match and the bottom symbol on the second reel is seven or bar.The condition for the second reel to stop early

Third reel

In mode bad or mode good, the third reel will always slip to prevent a disallowed match, i.e. any match in mode bad, or a match of sevens or bars in mode good. There is no upper limit for the distance the reel has to advance this way; as an extreme example, hacking the game so that the reels contain all sevens will likely cause the game to softlock.The piece of code run after finding a match
In addition, in mode good or mode super, the reel will slip up to 4 times if no match is found. This is not counting the number of slips done to skip disallowed seven or bar matches in mode good, so the total number of slips may exceed 4.The piece of code run after not finding a match
Notice that, due to the design of the reels, there can never be more than one match at any time.

Oddities

On the spin where the player enters the lucky streak, the actual mode stays as it is on the previous spin, which can be either mode bad or mode good (the mode is initialized to mode bad at the beginning of a sessionAll the memory addresses from $CD3D (wStoppingWhichSlotMachineWheel) to $CD50 (wSlotMachineBet) are zeroed, which included the mode flag at $CD4C.). If the mode happens to be mode good and the player wins, it will still count towards the 60 wins to end the lucky streak. There is a line of code to immediately end the lucky streak if the player wins a reward of 300 coins, but this seems to be impossible, because the lucky streak is mutually exclusive with mode super, and mode good doesn't allow matches of sevens. The probability distribution of the lucky slot machine is weird in multiple aspects. The fact that one machine is more likely to be lucky and some others have no chance might be intended, but the fact that it can be the nonexistent slot machine -1 seems certainly an off-by-one error. The fact that the first reel's early stop condition in mode super is never satisfied is most likely a bug. Presumably, the intention is that the reel should stop if a seven symbol is showing, but the developers mistakenly wrote jr c instead of jr z. The second reel's early stop condition in mode super is weird in two aspects. The 'bottom symbol on the second reel' condition seems like an accidental use of leftover variable, although it may also be intended since there are two instructions specifically making it the bottom symbolThese two dec de instructions make de point to the bottom symbol of the second reel instead of the top symbol, and they seem to serve no other purpose.. The 'sevens or bars' check is only for the first potential match found, in the order of 'bottom-bottom, bottom-middle, middle-middle, top-middle, top-top' (i.e. from bottom to top), which means a potential match of bars may be rerolled if it appears above a potential match of another symbol.

External links

Analysis of how the slot machines in Pokémon Red, Blue, Yellow work by Crystal_:REDIRECT Template:YouTube

References

Categories

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.

Slot Machine Probabilities

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.

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.

Slot Machine Probability Distribution Calculator

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

Slot Machines Probability

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.

Slot Machine Probability Distribution Rules

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.

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.

Slot Machine Probability Distribution Formula

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.

Slot Machine Probability Calculator

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.