1. Usual Slot Machine Probability Machine
2. Usual Slot Machine Probability Jackpots
3. Usual Slot Machine Probability Distribution
4. Slot Machine Probability Calculation

A slot machine house edge is known by casino managers as the “hold”, and hold percentages vary a great deal, and do tend to be smaller at more expensive slots, frequently found to be around 1- 3% at the five dollar slots.

Casino and gambling can be a lot of fun, especially when players do not understand the mathematics behind it, but being aware of these matters would offer players more fundamental understanding of the way probabilities and odds work.

Odds and probabilities are some of the most important aspects of gambling mathematics. They could be related even to the pay-off of a certain game. For example, there is an even money situation when it is explained that a bet pays off at the same odds as the probability of winning the bet.

Usual Slot Machine Probability Machine

Probabilities

Probabilities are often described as the very backbone of gambling mathematics.

Informally, probability is understood as the chance or the likelihood that a certain event will happen. It is also considered to be an estimate of the corresponding average frequency that is applied for a certain event to occur in a series of independent trials that have been repeated.

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The relative frequency of an event is somewhere between the probability of the event never to occur or always to occur, which basically means that it is always between 0% and 100%. Knowing how to use the probability would provide casino players with the opportunity to predict the eventual occurrence of an event. Unfortunately, it would not give them the chance to predict the exact moment when the event is to occur.

In addition, probability in gambling mathematics can be used when determining the conditions for acquiring certain results or even some financial expectations of a game in particular in the long term. Probabilities could also be helpful when determining how favourable a certain game would be and is it worth playing. This aspect of the matter is usually expressed as odds, as well as a fraction or a decimal fraction. Still, casino players should be aware of the fact that odds and probabilities are different things.

The estimate of the likelihood of a player winning divided by the total amount of available chances is called probability. The latter is also an ordinary fraction that can be expressed as a proportion between 0 and 1, or as a percentage.

What casino players should understand first is the formal theory of probability and are recommended to start with the term that is known as the “sample space”. It may sound somehow strange, but as a matter of fact, this is purely a description of all outcomes possible, or other words – of everything that can possibly happen. It is quite normal for sample spaces to be very large when it comes to casino games. As a matter of fact, they simply reflect the idea that too many things could actually happen to count them all.

The sample space is something that may occur in a large number of experiments and can normally be understood easily when considering the nature of the experiment. Still, the sample space does not have to be described in a explicit way.

Players will need be aware of some details in order to calculate the probability of a certain event. First, they will need a full count of the number of individual elements in the sample space. In addition, they will need to know the number of the individual elements related to the event. Once a player knows these values, they will be able to define the probability of the event by using the equation:

P(event) = Size of event / Size of the sample space

As shown in the equation, the letter P is used as a substitute of the word “probability”. What is not so evident by the above-mentioned equation is how the player should count the size of a variety of collections. Unfortunately, the counting problems in the gambling industry can be too complex. Of course, probabilities can be quickly calculated when counting is easy.

Odds

The most general definition of odds is the probability of a certain event happening in comparison to the probability of the same event not happening.

Odds are considered as ratios of the player's chances of losing to the likelihood of winning. In other words, they are the average frequency of a player's eventual loss to the average frequency of a win. As already mentioned above, odds are often mistaken with probabilities, but they are not the same thing.

As a matter of fact, odds almost never mean the actual likelihood of generating a win when it comes to gambling. In most cases, the term is used as a reference to a subjective estimate of the odds, and not as an exact mathematical calculation.

For example, if a player owns 1 of a total of 4 tickets, their probability to win is 1 in 4. Their odds, on the other hand, are 3 to 1. In order for a player to convert odds to probability, they need to take the likelihood for them to win, then use it as the numerator and divide by the total number of chances, including the ones for both winning and losing.

For example, let us take that the odds are 4 to 1. Then, the probability would be found by using the following equation:

1 / (1+4) = 1/5

In other words, the probability in this case would equal to 1/5, or 20%. What players need to know is the fact that the so-called “evens” are odds of 1 to 1. They are always paid even money.

The term “odds” could also be used as a substitute of the true odds that are the actual chances of winning. There are also payout odds, which represent the ratio of payout for each unit bet.

Just like probabilities, odds can also be expressed as a ratio of two numbers. The first number comes to represent the expected frequency of the occurrence of a specific outcome, and this is actually exactly what probabilities are. The second number, however, represents only the number of the rest of the possible outcomes. In comparison to probabilities, which could be expressed as percentages, odds are always displayed as ratios.

No, the odds are the same regardless of the time of day.

Most slot machines usually offer an incentive to play the maximum coins. For example two coins may pay 2,000 on the jackpot but three coins will pay 5,000. So if there an economy of scale incentive, then the return is higher with a max-coin bet. However, I should mention that most casinos tend to increase the theoretical return on their slots as they go up in denomination. So, you may be better off betting one coin on a $1 machine than four coins on a quarter machine.
My advice on slot selection is to play a simple smaller game. Nothing with fancy signage or a huge screen. Ultimately, it is the players that pay for that in the form of a lower return.

Interesting question. I'm sure that didn't refer to the payback percentage, 37% would be way too low. The 'hit frequency' is the probability that the player wins anything.

I'm pretty sure that the odds on Megabucks are the same everywhere. It is a 'proprietary game,' meaning the casino and slot maker (IGT) share in the profits. As I understand it, such propriety games are generally set to a return of about 88% by the slot maker, and the casinos do not have the option for a looser or tighter version.
The Wheel of Fortune game, with the big jackpot, I believe is also a proprietary game. Video poker odds are dictated by the pay table. For example, a 9-6 Jacks or Better game will pay 99.54%, assuming optimal strategy and an infinite amount of play, regardless of where the machine is or number of number of hands the player gets on the draw.

You're right, it is impossible for me to know without Microgaming giving me the details on how their reels are weighted. I have asked some of the major software companies for such information, but thus far nobody has volunteered anything. However, I can tell you that the average payback for all slots at the Golden Palace for the month of March 2000 was 95.67%. This information is available at the Golden Palace web site, click on the Price Waterhouse Coopers monthly payout review.

I would assume that the odds are the same at all Microgaming casinos. All casinos probably contribute money to the same account from which the jackpot is paid. This way, the individual casino from which the jackpot was hit does not have to reach into their own pocket when somebody wins. Mid-level payouts are probably paid by the casino itself.

The minimum applies to every machine. Someone with the Gaming Control Board in Carson City, Nevada, told me that every machine in the state must meet minimum payback percentages. The only exception, he said, are on some antique machines in Virginia City.

Unlike most slots, this game has different types of wins according to the number of coins bet. The first coin enables the player to win the small frequent 'bar' wins, from 2 to 60. The second coin enables larger 'seven' wins from 100 to 500. The third coin doubles the wins for sevens, except it also qualified the player for the progressive jackpot for three sizzling sevens.
The ways these games are programmed is to give the player a slightly higher return on each additional coin bet. For example, the first coin might have a return of 92%, the second 93%, and the third 94%. You seem to think the return for one coin would be very low, due to the small wins, but those wins happen more often than the wins for sevens.
In Nevada, regulations require slots to theoretically pay at least 75%. Even the games at the airport, which are very tight, still pay at least 85% or so. I'm quite sure that the return for any number of coins bet in Blazing Sevens conforms to industry norms.

In general Indian casinos are self-regulated. There is generally a tribal commission that will hear disputes, but ultimately the members of the commission know which side of their bread gets buttered.
Don't assume any kind of minimum return on the slot machines. However, ultimately economics would dictate that a return too low would be sensed by players, who would be unlikely to return if they consistently lost too much money too quickly. It would also be bad business, and time consuming, to loosen and tighten the slots like a yo-yo.
Your sound level hypothesis sounds interesting, I never thought of that.

The casinos don't actually program the casinos to pay a certain percentage, but determine the weighting of the reels so that the theoretical return is whatever they wish. In the short run, the actual return can be either much higher or lower than the theoretical return. However, the laws of mathematics dictate that the actual return will get closer to the theoretical return the greater the number of trials.

Each frame in these video slots is weighted equally. Any given line is equally likely to produce any given combination. Thus, the return is the same regardless of the number of coins played.

Royal flush - 840
Four deuces - 200
Wild Royal - 20
Five of a kind - 12
Straight flush - 9
Four of a kind - 5
Full house - 3
Flush - 2
Straight - 2
Three of a kind - 1
I would do this myself, but I am unable to use the necessary software, as I am not a windows user.

Physically all you have to do to change the return of the slot machine is change the EPROM chip inside. Assuming the casino manager had all the EPROM chips, which I think they sometimes do, they could make the change themselves. However in a major jurisdiction the change would have to be reported to the gaming authorities, not to mention internal paperwork. The random number generator is constant, it is what the program does with the random numbers that determines the return.

In general the higher the coinage the better the rate of return is. However in my own research I have seen plenty of exceptions, notably dollar machines that paid less than quarters.

Usual Slot Machine Probability Jackpots

I don’t know whether the chances of winning were better or not. They worked the same way as they do now except each stop on each reel had an equal chance. The very early ones didn’t pay money but chewing gum, which explains the bar symbols (sticks of gum) and fruits (flavors) on some modern slot machines.

I’ve been asked about these North Carolina slot machines so many times I’m tempted to fly there just to see them for myself. Yes, if they did give the probability of each symbol for each reel then an optimal strategy and a return could be fairly easily calculated. However I have never actually seen such a table and have never worked out the odds.

I just had an eye-opening experience at Casino Windsor. No where do they publish their percent payback on slots. However, that aside, I was going to play quarter (my comfort level) video poker. I was really taken aback when I put up the paytable. They were 5/4 machines. I am talking Jacks or better was only 5 coins on a full house and 4 coins on a flush. I looked at about 20 machines and only found one that paid any better and that was a 6/4 machine.

As I stated, I am not a mathematician but I think that payback percentage must be in the very low 70s. Needless to say I didn’t play video poker there because I know that the longer I would have played the greater the loss I could have expected with the house taking approximately $30.00 from every $100.00 put through the machine. That is not a gamble with some expectation of winning, it is a sure loss for players. On the Detroit side of the river, MGM Grand has their machines at 7/5. Not really great but a whole lot better than 5/4.

Could you please tell me the exact percentage payback on the 5/4 and 7/5 machines. Since none of the area casinos post their average payback on slots I am (and this is dangerous) assuming that their reel slots payback the same percentages. Best regards.

Actually with perfect play the 5/4 pay table return 92.78%. Still one of the worst pay tables I have ever heard of. Have you tried the Greektown casino in Detroit? I don't know what games they have but I do know they have had security remove several winning video poker players from the building, including a old lady who hit a royal on a machine with a 97% pay table. They must have something good enough to warrant throwing winners out for.

First lets clear up what the term 'hold' means. For purposes of electronic games it is the theoretical return the game is set to. In both video poker and slots each play is random and independent of all past plays. The laws of mathematics dictate that even with independent trials the as the sample size gets larger the actual return will tend to get closer to the theoretical mean, or the hold. So contrary to popular belief a machine never goes hot or cold to get back in balance. Never mind the term 'cycle.' It is a poorly named industry term for the number of possible outcomes of the random number generators inside the machine. Unfortunately the term has trickled down only to confuse low level employees and players alike. Contrary to popular myth there are no cycles and again each play is equally random and independent of all other plays.

Not often. Contrary to popular myth the casinos don’t tighten the machines on weekends or whenever it is busy. Here in Las Vegas the casinos have to fill out a form every time they change the percentage on each game. Most slot managers I have spoken with have a policy on what coinage is set to what return. I tend to think the most likely reason to change the percentage would a change in ownership and/or management, which do not happen often.

The 96.7% applies to total money bet and transaction fees generally only apply to deposits and/or withdrawals. Players generally circulate through the same money and thus bet much more than they deposit. As I discussed in the September 18, 2005, column a player could bet through about 1.5 million dollars with a $10,000 bankroll and betting $5 at a time in blackjack. In this case the casino would make their profit based on 1.5 million in bets but pay expenses based only on $10,000.

From what I hear anything you can configure at the machine you can configure remotely through the server. This would include the theoretical return percentage. However most casinos report that changing the theoretical return of a slot machine necessitates a lot of paperwork.

Even if it were effortless to move the slot machine return up and down, it strikes me as a conspiracy theory to think the casinos would do that on a player by player basis. Living here in Vegas, I hear all kinds of theories about the lengths the casinos go to in order to win, like pumping in oxygen and playing a subsonic mantra that says 'lose lose lose.' These are just urban legends. Most casinos correctly believe that if you give the player a good experience and a fair gamble then he will keep coming back. As they say, you can only slaughter a sheep only once, but you can shear it many times. (My webmaster, Michael Bluejay, who is a vegetarian, tried to get me to use this analogy instead: 'You can seize all of a sheep’s money only once, but you can force it to take you bowling many times.')

According to the Nevada Gaming Control Boards that means that every slot machine in that bank must be set to a theoretical return of at least 97.4%. Popular opinion has it that only one machine must be set to 97.4%, or that the overall average must be 97.4%. However, I am taking a firm stand that both notions are wrong, at least here in Nevada.

It depends. If the slot play may be used in any machine in the casino, including video poker, then it can be worth 99 cents on the dollar or more, depending on the video poker offerings. For example, the MGM Mirage casinos award $1 in free slot play for every one point earned. It can be used in any machine in any casino connected to the MGM Mirage player card. Most MGM Mirage properties offer 9/6 Jacks or better, so the value of $1 in free slot play is worth 99.54 cents, with correct strategy.

However, sometimes free slot play must be done on particular promotional machines that don’t accept money. The value of this kind of free play is hard to estimate closely, but generally very little. For example, the Las Vegas World used to sell “$1000” vacation packages for $400. Of the alleged $1000 value $600 was in promotional slot machine play. In his book “Million Dollar Video Poker,” Bob Dancer writes that he did this deal numerous times over, and estimates the value of the free slot play to be about ten cents on the dollar.

Usual Slot Machine Probability Distribution

It will show the specific return of the game you played.

Slot Machine Probability Calculation

Let’s assume 10.8 for the standard deviation, which I get from the Red, White, and Blue game described in my slot machine page. The standard deviation of the mean over n spins is standard deviation per bet divided by the square root of n. In this case, 10.8/10,000^0.5 = 0.108. The difference between 93% and 91% over 10,000 spins is just 18.5% of one standard deviation. To get the standard deviation of the mean to just 2% you would need a sample size of 291,600 spins. The standard deviation in slots will vary substantially, so take these figures with a grain of salt.

No, there isn’t. I don’t like it any more than you do. I think the player should be allowed to know the rules and/or the odds about what he is gambling on. Others have asked me if invoking the state Freedom of Information Act. I tend to doubt it would help or apply. As far as I know, the only place with such a right to know might be Holland. I’m told in Amsterdam information about the virtual reel stripping is indicated in little cards on the machines. You could in theory calculate the odds with that information and the pay table.

'Anonymous' .

No. To get an estimate of how loose a casino's slots are you would want to know the ratio of money returned to money bet. This ratio would typically be about 92%. Telling you only the amount paid out is not useful at all. Consider the Circus Circus claim of paying out $26 million a month. They don't disclose how much players bet to get that $26 million. What if they bet $35 million? That would be a return of 71.43%, which would be awful.
In conclusion, I don't know whether to be angry over what I consider to be deceptive advertising or depressed that people fall for it.