Slot games mathematics
Note: this article is based on microgaming slots - however the same logic applies to the full range of slot software makers. With this in mind I have put together a range of informative articles and slot playing guides that will help you select the perfect online slot game to play. These articles cover several different topics, which are based on several different aspects of each slot mentioned in them which have an effect on your overall winning chances.
Online slots math - understanding the numbers can help you choose the best slot games
Understanding the maths surrounding slot games
However, with many slot players becoming much more educated in regards to the way these types of gaming machines have been designed, and by getting a much better understanding of the mathematics surrounding the way they have been structured and designed, even a novice slot player can make a much more informed decision on which slot to play.
With this in mind I have put together a range of informative articles and slot playing guides that will help you select the perfect online slot game to play. These articles cover several different topics, which are based on several different aspects of each slot mentioned in them which have an effect on your overall winning chances.
Note: this article is based on microgaming slots - however the same logic applies to the full range of slot software makers.
While microgaming slots are not currently available at US friendly casinos, there are still a lot of excellent games that are. I recommend the bovada casino (part of the bodog family), where you will find slots from realtime gaming, rival gaming and betsoft side-by-side. Bovada also have some really cool player promos, check out www.Bovada.Lv for yourself!
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Should you enjoy playing slot games online at casinos powered by microgaming software, then these articles and slot playing guides are going to introduce you to a range of slots that have been designed to payout more than other slots, award their jackpots regularly and also trigger their respective bonus games more frequently.
We have also got a section dedicated to alerting you to the worst online slot games, by worst I mean those that have the lowest payout percentages or have been designed in such a way you are least likely to win a jackpot when playing them or slots that rarely trigger their bonus feature rounds!
Return to player payout percentages
One of the most confusing aspects of playing online slot games for many players is in regards to payouts percentages and slot game RTP's. Each slot game you will come across online will have been designed to return to players a proportion of the stakes wagered upon that game. Knowing what these figures are enables you to spot a slot which is going to give you higher winning payouts over the long term.
We have also put together guides that will show you how easy it is to work out your actual payout percentages of any slot session you play online, and this will enable you to keep quite detailed playing logs of just how well or how bad you are doing!
Bonus game feature rounds
Whether you like playing slots that can award you with sets of free spins, pick to win bonus games or even the popular spin the wheel type of bonus features, one article that we feel will be of interest to you is my how frequently are slot game bonus rounds triggered slot playing guide.
In it we list several microgaming slot games and take a look at how many scatter or bonus symbols are on each reel, and then work out what the actual chances are of you being awarded the bonus games offered on each of those slots.
You will be surprised at how many spins on average are actually required between bonus games being triggered. However, once you do find this information out, you can then play the slots which should award their respective bonus feature rounds more regularly, if of course bonus games are what excites you the most when playing online slots!
Frequency of slot jackpots being won
To enlighten you on what are the odds of winning a slot jackpot are, we have a range of slot games found in the gaming suite of microgaming software powered casinos listed, and have done all of the workings out for you, and as such present to you the odds of you spinning in the jackpot paying combination on all of those slots we have highlighted.
If you have not checked out the huge jackpots at microgaming slots recently, then you are in for a surprise! You'll regularly find the biggest ones at $2,000,000+ and plenty in the $100,000+ range are being won every week. Check out www.32red.Com for the latest totals!
Best paying online slot games
Each slot games found online will have been designed in such a way that players will get a certain percentage of their stakes returned to them as winnings. Each slot will have its own unique RTP which is the return to player payout percentage of that respective game.
The best slots to play are obviously those which through their design are set to give you more of your stakes back as payouts, and there are some huge differences in regards to just how much some slot games payback.
You will of course be seeking information on just which slots return more to players, and to help you in your quest to find the better paying slots we have the top paying slots currently found in all casinos utilizing microgaming’s suite of slot games.
Slot games best left alone
Even though slot games are completely random, if you do regularly play them online, then by opting to play only those slots offering you the best paybacks, you are ultimately going to have longer playing sessions. Plus a much better chance of winning than you would by playing slots which have been designed to return less to players via their respective RTP’s.
So to complement the above playing guide which showcases to you the best slots to play we have also compiled a guide of microgaming’s worst paying slot games. This guide will enlighten you fully on the slots you should never be tempted to play online, as those listed upon it have the lowest long term expected payout percentages, and as such offer very poor value and low winning opportunities, more so when you play slots online quite regularly!
Slot games mathematics
Configurations
there is a wide variety of the slot machines with regard to parametric design and rules. The configuration of a slot machine is specified by the configuration of its display and the configuration of its reels.
The display of a slot machine shows the outcomes of the reels in groups of spots (spot refers to a unit part of a reel holding one symbol, visible through its window; a spot on the display corresponds to a stop of the reel; a window can show one or more spots) having a certain shape and arrangement. The configuration of the display can be defined and modeled mathematically through a cartesian grid of integers, where the gridʼs points stand for the reel spots/stops and a (pay)line is a finite set of minimum 3 points that can be connected through a path linking successively neighboring points of that set. The length of a line is the cardinality of that set. Most of the slot machines have the display arranged as a rectangular grid. Lines can be of any shape and complexity and have all kinds of geometrical and topological properties. There are horizontal, vertical, oblique, or broken lines; symmetric, transversal lines; triangular, trapezoidal, zigzag, stair, or double-stair lines.
The distribution and arrangement of the symbols on each reel is also part of the configuration of a slot machine. On virtual slot machines and slot games in new slot sites, the physical configuration is replaced by parametric constraints upon the RNG (random number generator).
For the probability calculus in slots, only a part of the parameters and properties of the entire configuration of a slot machine do count [. ]. Read more on configurations
Winning combinations, slots events
any winning rule on a payline is expressed through a combination of symbols (for instance, the specific combination
With regard to the complexity of the events in respect to the ease of the probability computations, we have:
Simple events. These are the events related to one line, which are types of combinations of stops expressed through specific numbers of identical symbols (instances). For example,
Complex events of type 1. These are unions of simple events related to one line. For instance, the event any triple on a payline of length 3 of a fruit machine is a complex event of type 1, as being the union of the simple events
Complex events of type 2. These are events that are types of combinations of stops expressed through specific numbers of identical symbols, related to several lines. For instance,
Complex events of type 3. These are unions of events that are types of combinations of stops expressed through specific numbers of identical symbols (like the complex events of type 2), related to several lines. For instance, any triple on paylines 1 or 2 is a complex event of type 3. At least one cherry on at least one payline is also a complex event of type 3.
General formulas of the probability of the winning events related to one payline
For an event E related to a line of length n, the general formula of the probability of E is:
Where F(E) is the number of combinations of stops favorable for the event E to occur.
For an event E expressed through the number of instances of each symbol on a payline in case A, formula (1) is equivalent to:
Where
Formula (2) can be directly applied for winning events defined through the distribution of all symbols on the payline, in case A. These are simple events. For more complex events, we must apply the general formula (1), which reverts to counting the number of favorable combinations of stops F(E), or, for particular situations, apply formula (2) several times and add the results.
In case B, the number of variables is larger and therefore most of the explicit formulas from case B are too overloaded. We take here one particular type of events for which we present its probability formula in terms of basic probabilities, namely the events expressed through a number of instances of one symbol. If E is the event exactly m instances of S (
Where
Probability calculus tools for events related to several lines
for events related to several lines, other properties of probability are used (for instance, the inclusion-exclusion principle), along with formulas (1) and (2) and some approximation methods necessary for the ease of computations. When estimating the probability of an event related to several lines, some topological properties of that group of lines do count; for instance, the independence of the lines:
We call two lines independent if they do not contain stops of the same reel. This means that the outcome on one line does not depend on the outcome of the other and vice versa. Two lines that are not independent will be called non-independent.
For two non-independent lines, the outcome of one is influenced (partially or totally) by the outcome of the other. This definition can be extended to several lines (m), as follows: we call m lines independent if every pair of lines from them are independent. From probabilistic point of view, any two or more events each related to a line from a group of independent lines are independent, in the sense of the definition of independence of events from probability theory.
Independent and non-independent lines in a 3 x 3-display of a 9-reel slot machine
In the previous figure, lines
Non-independent lines in a 4 x 5-display of a 5-reel slot machine
In the previous figure, lines
An immediate consequence of the definition of independent lines is that if two lines intersect each other (that is, they share common stops), they are non-independent, so any group of lines containing them will be non-independent. Another consequence is that if two lines are independent, they do not intersect each other.
If two lines do not intersect each other, they are not necessarily independent. For instance, take lines
The non-independent lines (intersecting or non-intersecting) for which there are non-shared stops belonging to the same reels (like lines
All probabilities were worked out under the following assumptions:
- the reels spin independently;
- a payline does not contain two stops of the same reel (it crosses over the reels without overlapping them); this reverts to the fact that any m events, each one related to one stop of the payline, are independent of each other;
- each reel contains p symbols; this is actually a convention: if a symbol does not appear on a reel, we could simply take its distribution on that reel as being zero.
Given parameters
of course, any practical application can be fulfilled only if we know in advance the parameters of the given slot machine, that is, the numbers of stops of the reels and the symbol distributions on the reels. All the probability formulas and tables of values are ultimately useless without this information.
In the book the mathematics of slots: configurations, combinations, probabilities you will find explained some methods of estimating these parameters based on empirical data collected through statistical observation and physical measurements. Of course, taking into account the incomputable error ranges of such approximations, any credible information regarding these parameters should prevail over these methods of estimating them.
The mathematics department of infarom will launch soon the project probability sheet for any slot game, dealing with collecting statistical data from slot players, using the data to estimate the parameters of the slot machine, refining the estimations with the newly collected data and computing the probabilities and other statistical indicators attached to the payout schedule of the slot machine, in order to provide the so-called PAR sheet of any slot game on the market. Contact us with subject "slots data project" if you want to be part of our future project.
Practical applications and numerical probabilities
This section is dedicated to practical results, in which the general formulas are particularized in order to provide results for the most common categories of slot games and winning events. The practical results are presented as both specific formulas, ready for inputting the parameters of the slot game, and computed numerical results, where the specific formulas allow the generation of two-dimensional tables of values. The collection of results hold for winning combinations with no wild symbols (jokers) and is partial. You can find the complete collection of practical results in the book the mathematics of slots: configurations, combinations, probabilities, for 3-reel, 5-reel, 9-reel, and 16-reel slot machines.
3-reel slot machines
the 3-reel slot machines could have the following common configurations of the display: 1 x 3, 2 x 3, 3 x 3. The standard length of a payline is 3. The common winning events on a payline are:
(for example, (
(for example, (
(for example, (
(for example, (mix
(for example, (any bar any bar any bar ), with three bar symbols like
(the symbols from the examples are just for illustrating the winning combinations and may be replaced by symbols of any graphic. For the same parameters of the machine, the probabilities of the above events are the same regardless of the chosen graphic for the symbols.)
unions of winning events on a payline (disjunctions of the previous events
8. A specific symbol at least twice
9. A specific symbol at least once
10. A specific symbol three times or another specific symbol twice
11. A specific symbol three times or another specific symbol once
12. A specific symbol three times or another specific symbol at least once
13. A specific symbol three times or any combination of that symbol with another specific symbol
14. A specific symbol twice or another specific symbol once
15. A specific symbol twice or any combination of at least one of three other specific symbols
On a 3-reel 2 x 3- or 3 x 3-display slot machine, any two paylines are linked; therefore we cannot estimate the probabilities of the winning events related to several lines.
16-reel slot machines
the 16-reel slot machines usually have the 4 x 4 configuration of the display. The standard length of a payline is 4, but it could also have the length 3, 6, 7, or 8. The 16-reel 4 x 4-display slot machine could have 8 to 22 paylines of length 4, as follows: 4 horizontal, 4 vertical, 2 oblique (diagonal), or 12 trapezoidal lines. It could also have 4 transversal stair lines of length 7, 12 double-stair lines of length 6, or 10 double-stair lines of length 8. It could also have 4 oblique lines of length 3.
The common winning events on a payline are:
The table notes the probabilities of the winning events on a payline of length 4.
Unions of winning events on a payline (disjunctions of the previous events
7. A specific symbol at least three times
8. A specific symbol four times or another specific symbol three times
9. A specific symbol four times or another specific symbol at least three times
10. A specific symbol four times or any combination of that symbol with another specific symbol
11. A specific symbol three times or any combination of at least one of three other specific symbols
Winning events on several paylines
for the probabilities of these events, we considered only paylines of the regular length 4 in case A.
1.1 A winning event on any of the horizontal lines
1.2 A winning event on any of the vertical lines
1.3 A winning event on any of the horizontal or vertical lines
1.4 A winning event on either or both of the diagonals
1.5 A winning event on any of the horizontal or diagonal lines
1.6 A winning event on any of the vertical or diagonal lines
1.7 A winning event on any of the horizontal, vertical, or diagonal lines
1.8 A winning event on any of the left-right trapezoidal lines
1.9 A winning event on any of the horizontal or left-right trapezoidal lines
All slots probabilities and other statistical indicators, for the most common types of slot machines and the most common winning events, are covered in the book the mathematics of slots: configurations, combinations, probabilities . The collection of probability results is presented along with the mathematics behind the slot games. See the books section for details.
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Slots mathematics
Slots are immensely popular in both land-based and online casinos. They offer players a wide variety of exciting opportunities to win, with the progressive jackpot being the ultimate aim. As a player continues to play slots games, he/she could come across a host of various ways of winning, such as normal wins, scatter wins, holds and nudges, free spins, gambles and bonus games. In each of these there is a complex system of slots mathematics at play – and your chances of winning vary greatly according to what stage of the game is activated. For example, in bonus games you stand a much higher chance of winning than in normal play.
When we refer to the ‘chance of winning’ we draw heavily on the theory of probability, which is an essential part of gambling mathematics. To work out the probability of a particular win on a slots machine, you firstly need to use the combin funcion to calculate the number of ways that exist to form a particular combination, and then divide that figure by the total number of ways that exist to form any possible combination on the reels. For example, if you worked out that there are 250 ways to get a particular combination, and that there are a total of 250 000 possible combinations that a particular slots game could return, then the probability of your chosen combination coming up is 250/250 000 = 0.1%.
The payout percentage of slot machines
The payout percentage of slot machines is programmed into their electronically-programmable read-only memory (EPROM), and these figures can only be changed with proper authorization and approval from the relevant gaming authorities in a particular country/region. The correct procedures must also be followed to change the values for online slot machines. Slot machines typically have a payout percentage of between 85% and 98%, depending on the casino and the type of machine. The average house advantage for slots games has been calculated to be around the 9% mark.
Scaling returns
The slot machines in most land-based casinos operate on a system of scaling returns; that is, it’s more worth your while to play higher denominations, as the payout percentage of these machines tends to be higher than that of the lower denomination machines. In other words, higher denomination machines have a smaller house edge than the lower denomination machines. However, online casinos tend to keep their house edges – and thus payout percentages – consistent across all denominations.
Read about the casino game mathematics of popular games, or have a look at out gambling mathematics glossary.
Slot machine math: exploring game odds and hit frequency
Slot machines are very simple to play and don’t require any specific gambling knowledge. Anyone can get in the game. Every spin is just like every other spin – they’re all totally random. But just because they’re random doesn’t mean that there’s no math involved. All machines have one thing in common – the longer you play, the closer the payout will be to the theoretical results.
But in order to understand the theoretical odds it’s important to understand a little bit more about hit frequency and payback percentage.
Classic slots and classic odds
The classic mechanical slot machines used springs and tension to stop the reels at random. These old electro-mechanical slots weren’t anymore predictable than today’s modern slots. Most old mechanical machines had three reels that held 20 symbols, while most modern slots have three virtual reels with 22 stops – and a random number generator program.
For example, for an old slot machine with 20 stops on each reel there would be 8,000 different combinations. We get this number by multiplying the number of symbols on each reel (20 x 20 x 20 = 8,000). The 8,000 possible combinations are known as a cycle. It’s important to know that the cycle does not mean the cycle of winning and losing – just the mathematical cycle.
In theory, if you pulled the lever 8,000 times, you would see each combination of reels once. To figure out the odds of hitting a particular combination of three reels, simply multiply your chances of landing one symbol by itself (1/20 x 1/20 x 1/20). This means your odds of pulling up any combination on a classic slot machine are 1/8,000 or 7999 to 1.
Just like with flipping a coin – the odds may be 50/50, but you are not likely to see exactly 50 heads and exactly 50 tails if you flipped a coin 100 times. However, with both a slot machine or a coin flip, the higher number of times you do it, the closer the odds would be to the actually probability. So to actually get close to those odds, you’d have to flip the coins millions or billions of times.
How random number generators work
Modern slot machines use a random number generator (RNG). The RNG electronically generates a random value from millions of combinations. These values determine where the reel stops. Electronic slot machines generate these numbers at a rate of 300-500 per second. When you press the “spin” button on an electronic slot machine, it processes the random values. The first value it chooses determines the position of the first reel, the second value for the second reel and the third value for the third reel.
Most machines run a payback percentage somewhere between 75 and 99 percent. For example, if a slot machine has a 90percent payback, it will theoretically pay back 90 cents on every dollar paid in. Keep in mind that these percentages are figured from an infinite pool of spins.
Say you see an advertisement for a casino boasting a 95percent payback. What this is really saying is that “you put in $100 and we’ll give you back $95.” of course, you could win big on the first dollar or lose all $100. Think back on a random coin flip – you have a 50/50 chance of winning. In a totally just game where the casino has no advantage, you bet $100 and win you get $100 and if you lose you lose $100. Theoretically, over enough time, you and the casino would be even.
But with a 95 percent payback, the casino essentially pays you $95 for your win but still takes the $100 when you lose. In this scenario you are losing $5 for every $100 you wager. That’s how a casino profits – less winnings are paid than money gambled and lost.
The hit frequency basically tells you how many times you’ll have to lose in order to win – on average. A return of a couple of coins is a hit, but it’s not much of a win. For most slot machines, the hit frequency ranges from 9 to 25 percent. This turns out to a hit average of 9-25 times for every 100 spins conducted. These hits could be anything from a marginal return to a huge payout.
It’s important to note that there is usually no direct correlation between payout percentage and hit frequency. Machines with high payback may still have a low hit frequency or vice versa. A machine with a high payback and low hit frequency means that it doesn’t hit as often, but it pays out more. A machine with a low payback and high hit frequency pays out small wins, more often.
So whether you enjoy waiting longer for the bigger payout or getting little returns and less waiting time, you’ll now be able to find a slot machine that is the perfect combination of entertainment and profit.
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THE ULTIMATE GAMING MATHEMATICS GUIDE
The opportunity to play a casino game is a product that the consumer purchases with his gaming dollar. The view I take is that the player wants to have a certain experience when he sits at a game and is willing to pay for that experience by means of the inevitable bite the house edge takes.
To fully understand the product the player is purchasing requires a journey into the mathematics of casino games. Marketing must understand this material when developing effective strategies. Every part of a player’s interaction with a casino game has its own mathematics. Wagers have “odds” that determine how much to pay the player if he wins. The percentage of time the player wins is called the “hit frequency.” the “house edge” is the fraction of the player’s total initial wagers that the house expects to keep over the long run if the player uses perfect strategy. The “theo” or “theoretical” is the actual dollar amount the casino expects to win from the player. The “return to player” measures the fraction of the player’s total wagers the player is paid back in live casino play.
X-FOR-1, X-TO-1, 1-IN-X
Most table game wagers are paid in an "X-to-1" fashion. What this means is that if the player wins the round, his wager is returned and he is paid an additional amount for his win. If the player loses, then his wager is removed from the table.
For example, if the player bets on red-7 straight-up in roulette then the payout is 35-to-1. If his wager is $10, then he either wins $350 (35 × $10) and his $10 is returned, or he loses $10.
This type of pay is different from slot machines. For slots, every time the player pushes the button or pulls the handle, an amount is deducted from his credit balance. That's his wager. Then the reels spin, and when they stop he is paid an amount based on the pattern displayed by the reels by reference to some pay table. No matter what the outcome of the reels, the player’s original wager is forfeited. We say that slots are "X-for-1."
Most wagers on keno, lotteries, slots and video poker are paid "X-for-1.” this confusion has been used effectively, for example, in blackjack slot machines. Blackjack slots typically show a payout of 2 credits when the player is dealt a blackjack. The player knows that blackjack usually pays 3-to-2, so this seems like a sweet deal. However, this payout is 2-for-1. Thus the player forfeits his original 1 credit and then wins back 2 credits if he is dealt a blackjack. The player nets 1 credit in this way. Thus 2-for-1 is the same as 1-to-1. Few would play the table game of blackjack if the payout for a blackjack was reduced to 1-to-1, yet people often play these slots with a payout of 2-for-1.
The progressive wager in caribbean stud is an example of an "X-for-1" wager appearing in a well known table game. The player's wager is automatically collected as it is made, and a light illuminates on the table indicating the player has made this wager. Another common "X-for-1" wager is the bad-beat jackpot in the poker room. The pot contributes $1 each hand to the progressive; that money is forfeit.
It is customary that wagers pay out "X-to-1” on a table game; examples of successful “X-for-1 wagers on table games are rare (one example are the hop bets in craps). Return the player's original wager if the player wins or pushes; otherwise collect the player's wager. Table games and slots are different animals and the most fundamental difference is that slots pay “X-for-1.”
Some wagers in table games are very similar to slots. For example the pair plus wager in three card poker has a straight pay table based on the three cards dealt to the player. It usually pays “1-to-1” for a pair, “3-to-1” for a flush, “6-to-1” for a straight, “30-to-1” for trips and “40-to-1” for a straight flush. This side bet could just as well be a slot, as it requires no strategy, but it would then have the rather strange looking pay table: “2, 4, 7, 31, 41.”
Finally, the terminology "1-in-X" is commonly used, but not in the context of payouts. For example, in craps, the shooter will get a 7 (over the long run) 1-in-6 rolls. In double-0 roulette, the number red-7 will come up (over the long run) 1-in-38 spins. The player is dealt a blackjack (over the long run) approximately 1-in-21 hands. The terminology “1-in-X” has nothing to do with how much a game pays for a winning wager. It is used to describe the rate of occurrence of an event at a table game.
Figures & formulas: the maths behind slot games
Have you ever wondered about how casino slot payouts work? Well, wonder no more! Ladylucks casino have run the numbers and we can now explain how casinos generate their slot machine payouts.
We look at the difference between old mechanical slots and modern video slots, as well as going into detail about return to player rate (RTP) and slot volatility. We’ll also discuss how all these factors can ultimately affect your slots payouts. So buckle up and get ready for some maths! It’s not too difficult though – we promise.
Mechanical slots payouts
Mathematics and gambling have always been linked. Shrewd players hunt for ways to gain tiny advantages by utilising the facts and figures surrounding slot machine payouts, while casinos are searching for fresh mathematical tools that could help attract new players.
Slot machines were originally mechanical inventions. Let’s think of a classic three-reel fruit machine as our example, with only one payline. You win the top payout by landing three ‘lucky 7s’ in a row, where each reel has seven symbols in total. The chance of hitting the jackpot is 1 in (7x7x7), aka 1 in 343 or about 0.29%. In order to make money, the casino has to offer a jackpot lower than that – say £300 for betting £1. But if you’re used to modern day video slots, you’ll know that’s a pretty paltry top prize.
If casinos want to offer a bigger payout, they have to ensure the chances of landing the jackpot are as small as possible. One way to increase the jackpot on offer is to up the number of symbols on the reels. Changing from seven symbols to 12 per reel, decreases the slot machine payout odds to 1 in (12x12x12), which comes out to 1 in 1,728 or 0.06%. This allows the casino to set the jackpot at £1,700 – but it’s still not a huge amount of money, and has a very low chance of occurring.
Adapting the machine to a five-reel game adds a further two steps of multiplication to the calculation. Keeping the original seven-symbol reel, the chance of filling the payline with ‘lucky 7s’ suddenly jumps from 1 in 343 to 1 in 16,807 – so, in this case, the jackpot could reasonably increase to £16,000.
But if you think about the jackpots on offer from games today, they couldn’t utilise either of these methods. Imagine how many reels or symbols you’d need on a mechanical slot machine if you wanted to offer a jackpot worth hundreds of thousands!
But if you think about the jackpots on offer from games today, they couldn’t utilise either of these methods. Imagine how many reels or symbols you’d need on a mechanical slot machine if you wanted to offer a jackpot worth hundreds of thousands!
The origin of random slots
Another problem with mechanical slot machines was the impossibility of randomising them properly. Some casinos hooked them up to a timer which determined when they would pay out, but gamblers soon learned to exploit this. An example of this was professional blackjack and poker player cat hulbert: at one point, she employed teams of elderly contractors to play slots for her when they neared a payout time.
Slot machines were transformed in 1984 thanks to norwegian mathematician inge telnaes. His big idea was that, instead of spinning actual reels, a random number generator should dictate the results. You could create a visual representation of the slots and show either a winning combination or a random collection of losing symbols.
The random number generator could be programmed to offer odds that reels themselves can’t, and so you can offer much longer odds on winning combinations. IGT purchased telnaes’ patent back in 1989, and thus earned royalties as other companies started using the random number model too.
Whilst creating truly random numbers can be impossible, the pseudo-randomness that most random number generators offer is usually good enough. The strings of numbers which are generated can usually create long runs with many random properties. You can also manipulate the randomness of the numbers, though this must be accurately reflected by the return to player (RTP) listed on the machine.
How return to player rate works
Every slot you play online will have a RTP listed with it. This is typically somewhere between 92% and 98% and represents, in simulations, how much players will expect to receive back after an extended run of play.
In this case, the player would be set to walk away with £98 in their pocket. However, that’s far from the truth.
Slot machines are very rigorously tested and spun thousands upon thousands of times to ensure their advertised RTP is correct. If you ever feel like you’re not seeing the RTP advertised on them, it’s due to sheer randomness – and the weighting of jackpots, as you’ll see when we break down a slot in the next section.
A similar example to help wrap your head around this is flipping coins: you know that it’s a 50% chance to land heads or tails, but that’s over one flip. In 100 coin flips, you only actually have an 8% chance of having 50 heads and 50 tails.
Using binomial expansion, you can work out what the chance of this happening is – so, if you repeated the 100 flips 100 times, how many of those sets would actually have the perfect 50-50 split? The answer is 8%. But if you look at how many sets come out as close to 50% – say with between 40 to 60 heads – then the probability of this is much higher. The maths behind this is fairly complicated, but you can use a binomial calculator to prove it: the chance of success is 0.5, there are 100 trials, and 50 successes. You actually end up with a 7.96% of succeeding.
Use this as a reference for how unlikely perfect averages are: if you only get a perfect 50/50 split when flipping a coin 8% of the time, then think about how many different variations there are in a slot machine full of different symbols and stops. You would need a massive amount of time to see the average number overpower the inherent unpredictability of the random number generator.
Slot payouts’ percentage are worked out over a huge amount of time and a long string of spins. You need lots of spins to work out the average without being affected by the slot volatility. The longer you play a slot for, the nearer you will get to that elusive RTP figure – but your bankroll would have to be enormous to actually see it work out!
It’d be much more reasonable to think of rtps as expectations for millionaires: if you paid £1 million into a slot machine, you could expect to see £980,000 back from it. Obviously, no one person puts that in as one lump, but the machine might get that paid into it, in total, over the course of its lifetime. So the machine keeps £20,000 of the money that’s put in, assuming everything perfectly adheres to mathematical models.
Understanding slot payout : breaking down a simple slot
So now we know a bit about slots, let’s look at the maths behind the payouts. This is a very simple example, and it uses our three-reel, seven-symbol, one-payline slot that we outlined in the first section. For ease of argument, there are no special features here, no wild symbols, and no scatters. All of the paying combinations have to be found left-to-right in order to trigger the slot payout.
The slot payouts are going to be as follows:
- 3 lucky 7s = £100
- 3 of any other symbol = £10
- 2 lucky 7s = £10
- 2 of any other symbol = £3
There’s only one combination you can get to win the top prize, but we’ve introduced further methods of winning too. Three of any other symbol has six combinations, as does the chance of spinning two lucky 7s. But two of any other symbol is much more common, as there’s 36 ways that might show up in the slot machine.
Out of 343 combinations, only 49 in total (1+6+6+36) yield a reward, which only provides a 14.3% chance of pulling up a winning spin. However, that improbability is balanced out by higher wins.
If you multiply the number of combinations that can win by the number of coins you’d win, you come up with your potential pay-off per coin:
- 3 lucky 7s = 1 x £100 = 100
- 3 of any other symbol = 6 x £10 = 60
- 2 lucky 7s = 6 x £10 = 60
- 2 of any other symbol = 36 x £3 = 108
And adding these figures together gives us the total number of coins we could win from this spin
Divide this by the total number of spins – 343 – and times the result by 100, and we get the slot payout percentage:
With this slot, someone could come along and assume that putting £100 into this slot would give them a return of £95.60. However, looking at the maths behind it, we know that this isn’t necessarily true. It would be more accurate to think that a millionaire putting £1 million into it could walk away with £956,000 – and that the slot machine keeps the remaining £44,000 that it won through house edge.
What about slot volatility?
Slot volatility has a huge effect on how people play games. You can target different groups by changing a slot’s volatility up or down, though this might be represented by the same RTP.
A low volatility slot will reliably pay out small amounts over and over, and is ideal for players who want to keep spending their bankroll without too much chance of losing it all. However, high volatility slots are the ones with huge jackpots: they’re designed to draw you in for a quick shot at winning big money with the understanding it’s very unlikely.
In our lucky 7s slots, we can safely say that it has high volatility. Only 14.3% of spins will yield a winning combination; so, if you spin the slot 1,000 times, you’ll only be mathematically expected to trigger 143 wins. However, the draw of the jackpot – £100 for a single perfect combination – mitigates this. There’s still a high return on investment because of the significant boost that rolling a top prize gives.
Slots with many paylines and different distributions of symbols are much more complicated to calculate, but the principle remains the same. If big wins make up the bulk of the reward, it’s high volatility; if your slot payouts come from small and frequent wins, then it’s low volatility. And both high and low-volatility slots can vary in their advertised RTP.
Scaling returns
The final, minor point to make about slot machine maths is that some machines have scaling returns. If you ever see RTP shown as a range rather than as a single figure, it implies the slot you are playing on has different payout chances – and this usually depends on the amount of money you stake.
Consider a progressive slot game, where much of the RTP is made up of the small chance of landing a huge jackpot, there may be a higher chance of triggering the top prize if you stake more. This also means that you’re putting up more money to lose, but it does technically change the casino’s edge. Even if you only tilt the slot payout percentage a little bit – by 1% or 2% – the increased earnings on the slot can offset the larger wins, especially as many people may exhaust their bankroll into the machine without winning big money.
Scaling returns are a more common phenomenon in land-based casinos. Online casinos tend to be more consistent across all denominations, to ensure they don’t unfairly privilege high rollers.
Now you understand the maths behind slots, why not put your theoretical knowledge to the test? If you want to get the best slot payouts in the UK on online casinos, check out some of the great games available at ladylucks casino. We’ve got a great range of games you can enjoy, with a variety of paylines, slot volatility or RTP to suit every kind of player.
Slot games mathematics
Configurations
there is a wide variety of the slot machines with regard to parametric design and rules. The configuration of a slot machine is specified by the configuration of its display and the configuration of its reels.
The display of a slot machine shows the outcomes of the reels in groups of spots (spot refers to a unit part of a reel holding one symbol, visible through its window; a spot on the display corresponds to a stop of the reel; a window can show one or more spots) having a certain shape and arrangement. The configuration of the display can be defined and modeled mathematically through a cartesian grid of integers, where the gridʼs points stand for the reel spots/stops and a (pay)line is a finite set of minimum 3 points that can be connected through a path linking successively neighboring points of that set. The length of a line is the cardinality of that set. Most of the slot machines have the display arranged as a rectangular grid. Lines can be of any shape and complexity and have all kinds of geometrical and topological properties. There are horizontal, vertical, oblique, or broken lines; symmetric, transversal lines; triangular, trapezoidal, zigzag, stair, or double-stair lines.
The distribution and arrangement of the symbols on each reel is also part of the configuration of a slot machine. On virtual slot machines and slot games in new slot sites, the physical configuration is replaced by parametric constraints upon the RNG (random number generator).
For the probability calculus in slots, only a part of the parameters and properties of the entire configuration of a slot machine do count [. ]. Read more on configurations
Winning combinations, slots events
any winning rule on a payline is expressed through a combination of symbols (for instance, the specific combination
With regard to the complexity of the events in respect to the ease of the probability computations, we have:
Simple events. These are the events related to one line, which are types of combinations of stops expressed through specific numbers of identical symbols (instances). For example,
Complex events of type 1. These are unions of simple events related to one line. For instance, the event any triple on a payline of length 3 of a fruit machine is a complex event of type 1, as being the union of the simple events
Complex events of type 2. These are events that are types of combinations of stops expressed through specific numbers of identical symbols, related to several lines. For instance,
Complex events of type 3. These are unions of events that are types of combinations of stops expressed through specific numbers of identical symbols (like the complex events of type 2), related to several lines. For instance, any triple on paylines 1 or 2 is a complex event of type 3. At least one cherry on at least one payline is also a complex event of type 3.
General formulas of the probability of the winning events related to one payline
For an event E related to a line of length n, the general formula of the probability of E is:
Where F(E) is the number of combinations of stops favorable for the event E to occur.
For an event E expressed through the number of instances of each symbol on a payline in case A, formula (1) is equivalent to:
Where
Formula (2) can be directly applied for winning events defined through the distribution of all symbols on the payline, in case A. These are simple events. For more complex events, we must apply the general formula (1), which reverts to counting the number of favorable combinations of stops F(E), or, for particular situations, apply formula (2) several times and add the results.
In case B, the number of variables is larger and therefore most of the explicit formulas from case B are too overloaded. We take here one particular type of events for which we present its probability formula in terms of basic probabilities, namely the events expressed through a number of instances of one symbol. If E is the event exactly m instances of S (
Where
Probability calculus tools for events related to several lines
for events related to several lines, other properties of probability are used (for instance, the inclusion-exclusion principle), along with formulas (1) and (2) and some approximation methods necessary for the ease of computations. When estimating the probability of an event related to several lines, some topological properties of that group of lines do count; for instance, the independence of the lines:
We call two lines independent if they do not contain stops of the same reel. This means that the outcome on one line does not depend on the outcome of the other and vice versa. Two lines that are not independent will be called non-independent.
For two non-independent lines, the outcome of one is influenced (partially or totally) by the outcome of the other. This definition can be extended to several lines (m), as follows: we call m lines independent if every pair of lines from them are independent. From probabilistic point of view, any two or more events each related to a line from a group of independent lines are independent, in the sense of the definition of independence of events from probability theory.
Independent and non-independent lines in a 3 x 3-display of a 9-reel slot machine
In the previous figure, lines
Non-independent lines in a 4 x 5-display of a 5-reel slot machine
In the previous figure, lines
An immediate consequence of the definition of independent lines is that if two lines intersect each other (that is, they share common stops), they are non-independent, so any group of lines containing them will be non-independent. Another consequence is that if two lines are independent, they do not intersect each other.
If two lines do not intersect each other, they are not necessarily independent. For instance, take lines
The non-independent lines (intersecting or non-intersecting) for which there are non-shared stops belonging to the same reels (like lines
All probabilities were worked out under the following assumptions:
- the reels spin independently;
- a payline does not contain two stops of the same reel (it crosses over the reels without overlapping them); this reverts to the fact that any m events, each one related to one stop of the payline, are independent of each other;
- each reel contains p symbols; this is actually a convention: if a symbol does not appear on a reel, we could simply take its distribution on that reel as being zero.
Given parameters
of course, any practical application can be fulfilled only if we know in advance the parameters of the given slot machine, that is, the numbers of stops of the reels and the symbol distributions on the reels. All the probability formulas and tables of values are ultimately useless without this information.
In the book the mathematics of slots: configurations, combinations, probabilities you will find explained some methods of estimating these parameters based on empirical data collected through statistical observation and physical measurements. Of course, taking into account the incomputable error ranges of such approximations, any credible information regarding these parameters should prevail over these methods of estimating them.
The mathematics department of infarom will launch soon the project probability sheet for any slot game, dealing with collecting statistical data from slot players, using the data to estimate the parameters of the slot machine, refining the estimations with the newly collected data and computing the probabilities and other statistical indicators attached to the payout schedule of the slot machine, in order to provide the so-called PAR sheet of any slot game on the market. Contact us with subject "slots data project" if you want to be part of our future project.
Practical applications and numerical probabilities
This section is dedicated to practical results, in which the general formulas are particularized in order to provide results for the most common categories of slot games and winning events. The practical results are presented as both specific formulas, ready for inputting the parameters of the slot game, and computed numerical results, where the specific formulas allow the generation of two-dimensional tables of values. The collection of results hold for winning combinations with no wild symbols (jokers) and is partial. You can find the complete collection of practical results in the book the mathematics of slots: configurations, combinations, probabilities, for 3-reel, 5-reel, 9-reel, and 16-reel slot machines.
3-reel slot machines
the 3-reel slot machines could have the following common configurations of the display: 1 x 3, 2 x 3, 3 x 3. The standard length of a payline is 3. The common winning events on a payline are:
(for example, (
(for example, (
(for example, (
(for example, (mix
(for example, (any bar any bar any bar ), with three bar symbols like
(the symbols from the examples are just for illustrating the winning combinations and may be replaced by symbols of any graphic. For the same parameters of the machine, the probabilities of the above events are the same regardless of the chosen graphic for the symbols.)
unions of winning events on a payline (disjunctions of the previous events
8. A specific symbol at least twice
9. A specific symbol at least once
10. A specific symbol three times or another specific symbol twice
11. A specific symbol three times or another specific symbol once
12. A specific symbol three times or another specific symbol at least once
13. A specific symbol three times or any combination of that symbol with another specific symbol
14. A specific symbol twice or another specific symbol once
15. A specific symbol twice or any combination of at least one of three other specific symbols
On a 3-reel 2 x 3- or 3 x 3-display slot machine, any two paylines are linked; therefore we cannot estimate the probabilities of the winning events related to several lines.
16-reel slot machines
the 16-reel slot machines usually have the 4 x 4 configuration of the display. The standard length of a payline is 4, but it could also have the length 3, 6, 7, or 8. The 16-reel 4 x 4-display slot machine could have 8 to 22 paylines of length 4, as follows: 4 horizontal, 4 vertical, 2 oblique (diagonal), or 12 trapezoidal lines. It could also have 4 transversal stair lines of length 7, 12 double-stair lines of length 6, or 10 double-stair lines of length 8. It could also have 4 oblique lines of length 3.
The common winning events on a payline are:
The table notes the probabilities of the winning events on a payline of length 4.
Unions of winning events on a payline (disjunctions of the previous events
7. A specific symbol at least three times
8. A specific symbol four times or another specific symbol three times
9. A specific symbol four times or another specific symbol at least three times
10. A specific symbol four times or any combination of that symbol with another specific symbol
11. A specific symbol three times or any combination of at least one of three other specific symbols
Winning events on several paylines
for the probabilities of these events, we considered only paylines of the regular length 4 in case A.
1.1 A winning event on any of the horizontal lines
1.2 A winning event on any of the vertical lines
1.3 A winning event on any of the horizontal or vertical lines
1.4 A winning event on either or both of the diagonals
1.5 A winning event on any of the horizontal or diagonal lines
1.6 A winning event on any of the vertical or diagonal lines
1.7 A winning event on any of the horizontal, vertical, or diagonal lines
1.8 A winning event on any of the left-right trapezoidal lines
1.9 A winning event on any of the horizontal or left-right trapezoidal lines
All slots probabilities and other statistical indicators, for the most common types of slot machines and the most common winning events, are covered in the book the mathematics of slots: configurations, combinations, probabilities . The collection of probability results is presented along with the mathematics behind the slot games. See the books section for details.
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Slot machine math: exploring game odds and hit frequency
Slot machines are very simple to play and don’t require any specific gambling knowledge. Anyone can get in the game. Every spin is just like every other spin – they’re all totally random. But just because they’re random doesn’t mean that there’s no math involved. All machines have one thing in common – the longer you play, the closer the payout will be to the theoretical results.
But in order to understand the theoretical odds it’s important to understand a little bit more about hit frequency and payback percentage.
Classic slots and classic odds
The classic mechanical slot machines used springs and tension to stop the reels at random. These old electro-mechanical slots weren’t anymore predictable than today’s modern slots. Most old mechanical machines had three reels that held 20 symbols, while most modern slots have three virtual reels with 22 stops – and a random number generator program.
For example, for an old slot machine with 20 stops on each reel there would be 8,000 different combinations. We get this number by multiplying the number of symbols on each reel (20 x 20 x 20 = 8,000). The 8,000 possible combinations are known as a cycle. It’s important to know that the cycle does not mean the cycle of winning and losing – just the mathematical cycle.
In theory, if you pulled the lever 8,000 times, you would see each combination of reels once. To figure out the odds of hitting a particular combination of three reels, simply multiply your chances of landing one symbol by itself (1/20 x 1/20 x 1/20). This means your odds of pulling up any combination on a classic slot machine are 1/8,000 or 7999 to 1.
Just like with flipping a coin – the odds may be 50/50, but you are not likely to see exactly 50 heads and exactly 50 tails if you flipped a coin 100 times. However, with both a slot machine or a coin flip, the higher number of times you do it, the closer the odds would be to the actually probability. So to actually get close to those odds, you’d have to flip the coins millions or billions of times.
How random number generators work
Modern slot machines use a random number generator (RNG). The RNG electronically generates a random value from millions of combinations. These values determine where the reel stops. Electronic slot machines generate these numbers at a rate of 300-500 per second. When you press the “spin” button on an electronic slot machine, it processes the random values. The first value it chooses determines the position of the first reel, the second value for the second reel and the third value for the third reel.
Most machines run a payback percentage somewhere between 75 and 99 percent. For example, if a slot machine has a 90percent payback, it will theoretically pay back 90 cents on every dollar paid in. Keep in mind that these percentages are figured from an infinite pool of spins.
Say you see an advertisement for a casino boasting a 95percent payback. What this is really saying is that “you put in $100 and we’ll give you back $95.” of course, you could win big on the first dollar or lose all $100. Think back on a random coin flip – you have a 50/50 chance of winning. In a totally just game where the casino has no advantage, you bet $100 and win you get $100 and if you lose you lose $100. Theoretically, over enough time, you and the casino would be even.
But with a 95 percent payback, the casino essentially pays you $95 for your win but still takes the $100 when you lose. In this scenario you are losing $5 for every $100 you wager. That’s how a casino profits – less winnings are paid than money gambled and lost.
The hit frequency basically tells you how many times you’ll have to lose in order to win – on average. A return of a couple of coins is a hit, but it’s not much of a win. For most slot machines, the hit frequency ranges from 9 to 25 percent. This turns out to a hit average of 9-25 times for every 100 spins conducted. These hits could be anything from a marginal return to a huge payout.
It’s important to note that there is usually no direct correlation between payout percentage and hit frequency. Machines with high payback may still have a low hit frequency or vice versa. A machine with a high payback and low hit frequency means that it doesn’t hit as often, but it pays out more. A machine with a low payback and high hit frequency pays out small wins, more often.
So whether you enjoy waiting longer for the bigger payout or getting little returns and less waiting time, you’ll now be able to find a slot machine that is the perfect combination of entertainment and profit.
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Slot machines use maths to operate, so why can’t you use mathematics too beat them? Let’s fight fire with fire. Find below a math based approach to playing online slots at sites such as thorslots.Com.
Understanding the mathematics behind online slot games is invaluable in helping players choose the best machine. So let’s start by explaining how mathematics is used to operate slot machines.
Slots mathematics
The outcome of each spin on a slot machine is determined by a random number generator (RNG). These random number generators use mathematical formulas to generate random numbers, which correspond to the symbols on any given slot. If the random numbers generated correspond to a winning symbol combination, you win.
Calculating probabilities playing slots games
Probability refers to the likelihood of players winning at a slot machine. In order to calculate the probability of a specific slot, you need to know what a specific winning combination pays, as well as the probability of landing that combination.
How much any given combination pays out is usually listed in the paytable, but how do you calculate the probability of landing that combination? Easy. Just calculate the amount of stops in that slot. Let’s use a simple example. Let’s say that a slot contains 3 reels, with 5 symbols on each reel. To calculate the number of possible combinations on this machine, just multiply the number of symbols on each reel, by the number of stops on each reel. So this would be 5 x 5 x 5 = 125 possible winning combinations. If the jackpot in this slot pays out only for three cherries across the reels, and only one cherry occurs per reel, the calculation would be as follows: 1/5 x 1/5 x1/5 which equals a 0.008% chance at hitting the jackpot.
It is thus possible to calculate the probability of landing any combination as long as you know the number of times that each symbol appears on every reel, which is not information usually shared by online slots. Modern machines also come with many different numbers of reels and winning combinations, which could make it difficult to calculate its probability.
It is quite possible to calculate the probabilities of a simple and traditional slot machine. However, with modern slots it would be pretty much impossible, as slots can have as many as ten reels and 200 stops.
RTP percentages for online slot games
Return to player percentages (RTP) refers to the overall percentage of total wagers that a video slot game will pay out back to players over a period of time. Each online slot has a return to player percentage. For example, if a slot has an RTP of 98%, that means that for every dollar you bet, you will get back 98 cents. So to increase you odds of winning, you would want to choose the slot machines with the highest RTP rates, preferably 90+.
These are just a few of the ways in which you can use maths to choose the right online slot and improve your chances at winning.
Slot games mathematics
This guide, written by casino math professor robert hannum, contains a brief, non-technical discussion of the basic mathematics governing casino games and shows how casinos make money from these games. The article addresses a variety of topics, including house advantage, confusion about win rates, game volatility, player value and comp policies, casino pricing mistakes, and regulatory issues. Statistical advantages associated with the major games are also provided.
- Introduction
- Why is mathematics important?
- The house edge
- Probability versus odds
- Confusion about win rate
- Volatility and risk
- Player value and complimentaries
- Gaming regulation and mathematics
- Summary tables for house advantage
At its core the business of casino gaming is pretty simple. Casinos make money on their games because of the mathematics behind the games. As nico zographos, dealer-extraordinaire for the 'greek syndicate' in deauville, cannes, and monte carlo in the 1920s observed about casino gaming: "there is no such thing as luck. It is all mathematics."
With a few notable exceptions, the house always wins - in the long run - because of the mathematical advantage the casino enjoys over the player. That is what mario puzo was referring to in his famous novel fools die when his fictional casino boss character, gronevelt, commented: "percentages never lie. We built all these hotels on percentages. We stay rich on the percentage. You can lose faith in everything, religion and god, women and love, good and evil, war and peace. You name it. But the percentage will always stand fast."
Puzo is, of course, right on the money about casino gaming. Without the "edge," casinos would not exist. With this edge, and because of a famous mathematical result called the law of large numbers, a casino is guaranteed to win in the long run.
Why is mathematics important?
Critics of the gaming industry have long accused it of creating the name "gaming" and using this as more politically correct than calling itself the "gambling industry." the term "gaming," however, has been around for centuries and more accurately describes the operators' view of the industry because most often casino operators are not gambling. Instead, they rely on mathematical principles to assure that their establishment generates positive gross gaming revenues. The operator, however, must assure the gaming revenues are sufficient to cover deductions like bad debts, expenses, employees, taxes and interest.
Despite the obvious, many casino professionals limit their advancements by failing to understand the basic mathematics of the games and their relationships to casino profitability. One casino owner would often test his pit bosses by asking how a casino could make money on blackjack if the outcome is determined simply by whether the player or the dealer came closest to 21. The answer, typically, was because the casino maintained "a house advantage." this was fair enough, but many could not identify the amount of that advantage or what aspect of the game created the advantage. Given that products offered by casinos are games, managers must understand why the games provide the expected revenues. In the gaming industry, nothing plays a more important role than mathematics.
Mathematics should also overcome the dangers of superstitions. An owner of a major las vegas strip casino once experienced a streak of losing substantial amounts of money to a few "high rollers." he did not attribute this losing streak to normal volatility in the games, but to bad luck. His solution was simple. He spent the evening spreading salt throughout the casino to ward off the bad spirits. Before attributing this example to the idiosyncrasies of one owner, his are atypical only in their extreme. Superstition has long been a part of gambling - from both sides of the table. Superstitions can lead to irrational decisions that may hurt casino profits. For example, believing that a particular dealer is unlucky against a particular (winning) player may lead to a decision to change dealers. As many, if not most, players are superstitious. At best, he may resent that the casino is trying to change his luck. At worst, the player may feel the new dealer is skilled in methods to "cool" the game. Perhaps he is even familiar with stories of old where casinos employed dealers to cheat "lucky" players.
Understanding the mathematics of a game also is important for the casino operator to ensure that the reasonable expectations of the players are met. For most persons, gambling is entertainment. It provides an outlet for adult play. As such, persons have the opportunity for a pleasant diversion from ordinary life and from societal and personal pressures. As an entertainment alternative, however, players may consider the value of the gambling experience. For example, some people may have the option of either spending a hundred dollars during an evening by going to a professional basketball game or at a licensed casino. If the house advantage is too strong and the person loses his money too quickly, he may not value that casino entertainment experience. On the other hand, if a casino can entertain him for an evening, and he enjoys a "complimentary" meal or drinks, he may want to repeat the experience, even over a professional basketball game. Likewise, new casino games themselves may succeed or fail based on player expectations. In recent years, casinos have debuted a variety of new games that attempt to garner player interest and keep their attention. Regardless of whether a game is fun or interesting to play, most often a player will not want to play games where his money is lost too quickly or where he has a exceptionally remote chance of returning home with winnings.
Mathematics also plays an important part in meeting players' expectations as to the possible consequences of his gambling activities. If gambling involves rational decision-making, it would appear irrational to wager money where your opponent has a better chance of winning than you do. Adam smith suggested that all gambling, where the operator has an advantage, is irrational. He wrote "there is not, however, a more certain proposition in mathematics than that the more tickets [in a lottery] you advertise upon, the more likely you are a loser. Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets, the nearer you approach to this certainty."
Even where the house has an advantage, however, a gambler may be justified if the amount lost means little to him, but the potential gain would elevate him to a higher standing of living. For example, a person with an annual income of $30,000 may have $5 in disposable weekly income. He could save or gamble this money. By saving it, at the end of a year, he would have $260. Even if he did this for years, the savings would not elevate his economic status to another level. As an alternative, he could use the $5 to gamble for the chance to win $1 million. While the odds of winning are remote, it may provide the only opportunity to move to a higher economic class.
Since the casino industry is heavily regulated and some of the standards set forth by regulatory bodies involve mathematically related issues, casino managers also should understand the mathematical aspects relating to gaming regulation. Gaming regulation is principally dedicated to assuring that the games offered in the casino are fair, honest, and that players get paid if they win. Fairness is often expressed in the regulations as either requiring a minimum payback to the player or, in more extreme cases, as dictating the actual rules of the games offered. Casino executives should understand the impact that rules changes have on the payback to players to assure they meet regulatory standards. Equally important, casino executives should understand how government mandated rules would impact their gaming revenues.
The player's chances of winning in a casino game and the rate at which he wins or loses money depends on the game, the rules in effect for that game, and for some games his level of skill. The amount of money the player can expect to win or lose in the long run - if the bet is made over and over again - is called the player's wager expected value (EV), or expectation. When the player's wager expectation is negative, he will lose money in the long run. For a $5 bet on the color red in roulette, for example, the expectation is -$0.263. On the average the player will lose just over a quarter for each $5 bet on red.
When the wager expectation is viewed from the casino's perspective (i.E., the negative of the player's expectation) and expressed as a percentage, you have the house advantage. For the roulette example, the house advantage is 5.26% ($0.263 divided by $5). The formal calculation is as follows:
EV = (+5)(18/38) + (-5)(20/38) = -0.263
(house advantage = 0.263/5 = 5.26%)
When this EV calculation is performed for a 1-unit amount, the negative of the resulting value is the house edge. Here are the calculations for bets on a single-number in double-zero and single-zero roulette.
Double-zero roulette (single number bet):
EV = (+35)(1/38) + (-1)(37/38) = -0.053
(house advantage = 5.3%)
Single-zero roulette (single number bet):
EV = (+35)(1/37) + (-1)(36/37) = -0.027
(house advantage = 2.7%)
The house advantage represents the long run percentage of the wagered money that will be retained by the casino. It is also called the house edge, the "odds" (i.E., avoid games with bad odds), or just the "percentage" (as in mario puzo's fools die). Although the house edge can be computed easily for some games - for example, roulette and craps - for others it requires more sophisticated mathematical analysis and/or computer simulations. Regardless of the method used to compute it, the house advantage represents the price to the player of playing the game.
Because this positive house edge exists for virtually all bets in a casino (ignoring the poker room and sports book where a few professionals can make a living), gamblers are faced with an uphill and, in the long run, losing battle. There are some exceptions. The odds bet in craps has zero house edge (although this bet cannot be made without making another negative expectation wager) and there are a few video poker machines that return greater than 100% if played with perfect strategy. Occasionally the casino will even offer a promotion that gives the astute player a positive expectation. These promotions are usually mistakes - sometimes casinos don't check the math - and are terminated once the casino realizes the player has the edge. But by and large the player will lose money in the long run, and the house edge is a measure of how fast the money will be lost. A player betting in a game with a 4% house advantage will tend to lose his money twice as fast as a player making bets with a 2% house edge. The trick to intelligent casino gambling - at least from the mathematical expectation point of view - is to avoid the games and bets with the large house advantages.
Some casino games are pure chance - no amount of skill or strategy can alter the odds. These games include roulette, craps, baccarat, keno, the big-six wheel of fortune, and slot machines. Of these, baccarat and craps offer the best odds, with house advantages of 1.2% and less than 1% (assuming only pass/come with full odds), respectively. Roulette and slots cost the player more - house advantages of 5.3% for double-zero roulette and 5% to 10% for slots - while the wheel of fortune feeds the casino near 20% of the wagers, and keno is a veritable casino cash cow with average house advantage close to 30%.
Games where an element of skill can affect the house advantage include blackjack, video poker, and the four popular poker-based table games: caribbean stud poker, let it ride, three card poker, and pai gow poker. For the poker games, optimal strategy results in a house edge in the 3% to 5% range (CSP has the largest house edge, PGP the lowest, with LIR and TCP in between). For video poker the statistical advantage varies depending on the particular machine, but generally this game can be very player friendly - house edge less than 3% is not uncommon and some are less than 1% - if played with expert strategy.
Blackjack, the most popular of all table games, offers the skilled player some of the best odds in the casino. The house advantage varies slightly depending on the rules and number of decks, but a player using basic strategy faces little or no disadvantage in a single-deck game and only a 0.5% house edge in the common six-deck game. Despite these numbers, the average player ends up giving the casino a 2% edge due to mistakes and deviations from basic strategy. Complete basic strategy tables can be found in many books and many casino-hotel gift shops sell color-coded credit card size versions. Rule variations favorable to the player include fewer decks, dealer stands on soft seventeen (worth 0.2%), doubling after splitting (0.14%), late surrender (worth 0.06%), and early surrender (uncommon, but worth 0.24%). If the dealer hits soft seventeen it will cost you, as will any restrictions on when you can double down.
Probability represents the long run ratio of (# of times an outcome occurs) to (# of times experiment is conducted). Odds represent the long run ratio of (# of times an outcome does not occur) to (# of times an outcome occurs). If a card is randomly selected from a standard deck of 52 playing cards, the probability it is a spade is 1/4; the odds (against spade) are 3 to 1. The true odds of an event represent the payoff that would make the bet on that event fair. For example, a bet on a single number in double-zero roulette has probability of 1/38, so to break even in the long run a player would have to be paid 37 to 1 (the actual payoff is 35 to 1).
There are all kinds of percentages in the world of gaming. Win percentage, theoretical win percentage, hold percentage, and house advantage come to mind. Sometimes casino bosses use these percentages interchangeably, as if they are just different names for the same thing. Admittedly, in some cases this is correct. House advantage is just another name for theoretical win percentage, and for slot machines, hold percentage is (in principle) equivalent to win percentage. But there are fundamental differences among these win rate measurements.
The house advantage - the all-important percentage that explains how casinos make money - is also called the house edge, the theoretical win percentage, and expected win percentage. In double-zero roulette, this figure is 5.3%. In the long run the house will retain 5.3% of the money wagered. In the short term, of course, the actual win percentage will differ from the theoretical win percentage (the magnitude of this deviation can be predicted from statistical theory). The actual win percentage is just the (actual) win divided by the handle. Because of the law of large numbers - or as some prefer to call it, the law of averages - as the number of trials gets larger, the actual win percentage should get closer to the theoretical win percentage.
Because handle can be difficult to measure for table games, performance is often measured by hold percentage (and sometimes erroneously called win percentage). Hold percentage is equal to win divided by drop. In nevada, this figure is about 24% for roulette. The drop and hold percentage are affected by many factors; we won't delve into these nor the associated management issues. Suffice it to say that the casino will not in the long term keep 24% of the money bet on the spins of roulette wheel - well, an honest casino won't.
To summarize: house advantage and theoretical win percentage are the same thing, hold percentage is win over drop, win percentage is win over handle, win percentage approaches the house advantage as the number of plays increases, and hold percentage is equivalent to win percentage for slots but not table games.
· hold % = win/drop
· win % (actual) = win/handle
· H.A. = theoretical win % = limit(actual win %) = limit(win/handle)
· hold percentage ¹ house edge
Furthermore, the house advantage is itself subject to varying interpretations. In let it ride, for example, the casino advantage is either 3.51% or 2.86% depending on whether you express the advantage with respect to the base bet or the average bet. Those familiar with the game know that the player begins with three equal base bets, but may withdraw one or two of these initial units. The final amount put at risk, then, can be one (84.6% of the time assuming proper strategy), two (8.5%), or three units (6.9%), making the average bet size 1.224 units. In the long run, the casino will win 3.51% of the hands, which equates to 2.86% of the money wagered. So what's the house edge for let it ride? Some prefer to say 3.51% per hand, others 2.86% per unit wagered. No matter. Either way, the bottom line is the same either way: assuming three $1 base bets, the casino can expect to earn 3.5¢ per hand (note that 1.224 x 0.0286 = 0.035).
The question of whether to use the base bet or average bet size also arises in caribbean stud poker (5.22% vs. 2.56%), three card poker (3.37% vs. 2.01%), casino war (2.88% vs. 2.68%), and red dog (2.80% vs. 2.37%).
For still other games, the house edge can be stated including or excluding ties. The prime examples here are the player (1.24% vs. 1.37%) and banker (1.06% vs. 1.17%) bets in baccarat, and the don't pass bet (1.36% vs. 1.40%) in craps. Again, these are different views on the casino edge, but the expected revenue will not change.
That the house advantage can appear in different disguises might be unsettling. When properly computed and interpreted, however, regardless of which representation is chosen, the same truth (read: money) emerges: expected win is the same.
Statistical theory can be used to predict the magnitude of the difference between the actual win percentage and the theoretical win percentage for a given number of wagers. When observing the actual win percentage a player (or casino) may experience, how much variation from theoretical win can be expected? What is a normal fluctuation? The basis for the analysis of such volatility questions is a statistical measure called the standard deviation (essentially the average deviation of all possible outcomes from the expected). Together with the central limit theorem (a form of the law of large numbers), the standard deviation (SD) can be used to determine confidence limits with the following volatility guidelines:
Volatility analysis guidelines
· only 5% of the time will outcomes will be more than 2 SD's from expected outcome
· almost never (0.3%) will outcomes be more than 3 SD's from expected outcome
Obviously a key to using these guidelines is the value of the SD. Computing the SD value is beyond the scope of this article, but to get an idea behind confidence limits, consider a series of 1,000 pass line wagers in craps. Since each wager has a 1.4% house advantage, on average the player will be behind by 14 units. It can be shown (calculations omitted) that the wager standard deviation is for a single pass line bet is 1.0, and for 1,000 wagers the SD is 31.6. Applying the volatility guidelines, we can say that there is a 95% chance the player's actual win will be between 49 units ahead and 77 units behind, and almost certainly between 81 units ahead and 109 units behind.
A similar analysis for 1,000 single-number wagers on double-zero roulette (on average the player will be behind 53 units, wager SD = 5.8, 1,000 wager SD = 182.2) will yield 95% confidence limits on the player win of 311 units ahead and 417 units behind, with win almost certainly between 494 units ahead and 600 units behind.
Note that if the volatility analysis is done in terms of the percentage win (rather than the number of units or amount won), the confidence limits will converge to the house advantage as the number of wagers increases. This is the result of the law of large numbers - as the number of trials gets larger, the actual win percentage should get closer to the theoretical win percentage. Risk in the gaming business depends on the house advantage, standard deviation, bet size, and length of play.
Player value and complimentaries
Using the house advantage, bet size, duration of play, and pace of the game, a casino can determine how much it expects to win from a certain player. This player earning potential (also called player value, player worth, or theoretical win) can be calculated by the formula:
Earning potential = average bet ´ hours played ´ decisions per hour ´ house advantage
For example, suppose a baccarat player bets $500 per hand for 12 hours at 60 hands per hour. Using a house advantage of 1.2%, this player's worth to the casino is $4,320 (500 ´ 12 ´ 60 ´ .012). A player who bets $500 per spin for 12 hours in double-zero roulette at 60 spins per hour would be worth about $19,000 (500 ´ 12 ´ 60 ´ .053).
Many casinos set comp (complimentary) policies by giving the player back a set percentage of their earning potential. Although comp and rebate policies based on theoretical loss are the most popular, rebates on actual losses and dead chip programs are also used in some casinos. Some programs involve a mix of systems. The mathematics associated with these programs will not be addressed in this article.
In an effort to entice players and increase business, casinos occasionally offer novel wagers, side bets, increased payoffs, or rule variations. These promotions have the effect of lowering the house advantage and the effective price of the game for the player. This is sound reasoning from a marketing standpoint, but can be disastrous for the casino if care is not taken to ensure the math behind the promotion is sound. One casino offered a baccarat commission on winning banker bets of only 2% instead of the usual 5%, resulting in a 0.32% player advantage. This is easy to see (using the well-known probabilities of winning and losing the banker bet):
EV = (+0.98)(.4462) + (-1)(.4586) = 0.0032
(house advantage = -0.32%)
A casino in biloxi, mississippi gave players a 12.5% edge on sic bo bets of 4 and 17 when they offered 80 to 1 payoffs instead of the usual 60 to 1. Again, this is an easy calculation. Using the fact that the probability of rolling a total of 4 (same calculation applies for a total of 17) with three dice is 1/72 (1/6 x 1/6 x 1/6 x 3), here are the expected values for both the usual and the promotional payoffs:
Usual 60 to 1 payoff: EV = (+60)(1/72) + (-1)(71/72) = -0.153
(house advantage = 15.3%)
Promotional 80 to 1 payoff: EV = (+80)(1/72) + (-1)(71/72) = +0.125
(house advantage = -12.5%)
In other promotional gaffes, an illinois riverboat casino lost a reported $200,000 in one day with their "2 to 1 tuesdays" that paid players 2 to 1 (the usual payoff is 3 to 2) on blackjack naturals, a scheme that gave players a 2% advantage. Not to be outdone, an indian casino in california paid 3 to 1 on naturals during their "happy hour," offered three times a day, two days a week for over two weeks. This promotion gave the player a whopping 6% edge. A small las vegas casino offered a blackjack rule variation called the "free ride" in which players were given a free right-to-surrender token every time they received a natural. Proper use of the token led to a player edge of 1.3%, and the casino lost an estimated $17,000 in eight hours. Another major las vegas casino offered a "50/50 split" blackjack side bet that allowed the player to stand on an initial holding of 12-16, and begin a new hand for equal stakes against the same dealer up card. Although the game marketers claimed the variation was to the advantage of the casino, it turned out that players who exercised the 50/50 split only against dealer 2-6 had a 2% advantage. According to one pit boss, the casino suffered a $230,000 loss in three and a half days.
In the gaming business, it's all about "bad math" or "good math." honest games based on good math with positive house advantage minimize the short-term risk and ensure the casino will make money in the long run. Players will get "lucky" in the short term, but that is all part of the grand design. Fluctuations in both directions will occur. We call these fluctuations good luck or bad luck depending on the direction of the fluctuation. There is no such thing as luck. It is all mathematics.
Gaming regulation and mathematics
Casino gaming is one of the most regulated industries in the world. Most gaming regulatory systems share common objectives: keep the games fair and honest and assure that players are paid if they win. Fairness and honesty are different concepts. A casino can be honest but not fair. Honesty refers to whether the casino offers games whose chance elements are random. Fairness refers to the game advantage - how much of each dollar wagered should the casino be able to keep? A slot machine that holds, on average, 90% of every dollar bet is certainly not fair, but could very well be honest (if the outcomes of each play are not predetermined in the casino's favor). Two major regulatory issues relating to fairness and honesty - ensuring random outcomes and controlling the house advantage - are inextricably tied to mathematics and most regulatory bodies require some type of mathematical analysis to demonstrate game advantage and/or confirm that games outcomes are random. Such evidence can range from straightforward probability analyses to computer simulations and complex statistical studies. Requirements vary across jurisdictions, but it is not uncommon to see technical language in gaming regulations concerning specific statistical tests that must be performed, confidence limits that must be met, and other mathematical specifications and standards relating to game outcomes.
Summary tables for house advantage
The two tables below show the house advantages for many of the popular casino games. The first table is a summary of the popular games and the second gives a more detailed breakdown.
| House advantages for popular casino games | |
| game | house advantage |
| roulette (double-zero) | 5.3% |
| craps (pass/come) | 1.4% |
| craps (pass/come with double odds) | 0.6% |
| blackjack - average player | 2.0% |
| blackjack - 6 decks, basic strategy* | 0.5% |
| blackjack - single deck, basic strategy* | 0.0% |
| baccarat (no tie bets) | 1.2% |
| caribbean stud* | 5.2% |
| let it ride* | 3.5% |
| three card poker* | 3.4% |
| pai gow poker (ante/play)* | 2.5% |
| slots | 5% - 10% |
| video poker* | 0.5% - 3% |
| keno (average) | 27.0% |
| *optimal strategy | |
| House advantages for major casino wagers | ||
| game | bet | HA* |
| baccarat | banker (5% commission) | 1.06% |
| baccarat | player | 1.24% |
| big six wheel | average | 19.84% |
| blackjack | card-counting | -1.00% |
| blackjack | basic strategy | 0.50% |
| blackjack | average player | 2.00% |
| blackjack | poor player | 4.00% |
| caribbean stud | ante | 5.22% |
| casino war | basic bet | 2.88% |
| craps | any craps | 11.11% |
| craps | any seven | 16.67% |
| craps | big 6, big 8 | 9.09% |
| craps | buy (any) | 4.76% |
| craps | C&E | 11.11% |
| craps | don't pass/don't come | 1.36% |
| craps | don't pass/don't come w/1X odds | 0.68% |
| craps | don't pass/don't come w/2X odds | 0.45% |
| craps | don't pass/don't come w/3X odds | 0.34% |
| craps | don't pass/don't come w/5X odds | 0.23% |
| craps | don't pass/don't come w/10X odds | 0.12% |
| craps | don't place 4 or 10 | 3.03% |
| craps | don't place 5 or 9 | 2.50% |
| craps | don't place 6 or 8 | 1.82% |
| craps | field (2 and 12 pay double) | 5.56% |
| craps | field (2 or 12 pays triple) | 2.78% |
| craps | hard 4, hard 10 | 11.11% |
| craps | hard 6, hard 8 | 9.09% |
| craps | hop bet - easy (14-1) | 16.67% |
| craps | hop bet - easy (15-1) | 11.11% |
| craps | hop bet - hard (29-1) | 16.67% |
| craps | hop bet - hard (30-1) | 13.89% |
| craps | horn bet (30-1 & 15-1) | 12.50% |
| craps | horn high - any (29-1 & 14-1) | 16.67% |
| craps | horn high 2, horn high 12 (30-1 & 15-1) | 12.78% |
| craps | horn high 3, horn high 11 (30-1 & 15-1) | 12.22% |
| craps | lay 4 or 10 | 2.44% |
| craps | lay 5 or 9 | 3.23% |
| craps | lay 6 or 8 | 4.00% |
| craps | pass/come | 1.41% |
| craps | pass/come w/1X odds | 0.85% |
| craps | pass/come w/2X odds | 0.61% |
| craps | pass/come w/3X odds | 0.47% |
| craps | pass/come w/5X odds | 0.33% |
| craps | pass/come w/10X odds | 0.18% |
| craps | place 4 or 10 | 6.67% |
| craps | place 5 or 9 | 4.00% |
| craps | place 6 or 8 | 1.52% |
| craps | three, eleven (14-1) | 16.67% |
| craps | three, eleven (15-1) | 11.11% |
| craps | two, twelve (29-1) | 16.67% |
| craps | two, twelve (30-1) | 13.89% |
| keno | typical | 27.00% |
| let it ride | base bet | 3.51% |
| pai gow | poker skilled player (non-banker) | 2.54% |
| pai gow poker | average player (non-banker) | 2.84% |
| red dog | basic bet (six decks) | 2.80% |
| roulette | single-zero | 2.70% |
| roulette | double-zero (except five-number) | 5.26% |
| roulette | double-zero, five-number bet | 7.89% |
| sic bo | big/small | 2.78% |
| sic bo | one of a kind | 7.87% |
| sic bo | 7, 14 | 9.72% |
| sic bo | 8, 13 | 12.50% |
| sic bo | 10, 11 | 12.50% |
| sic bo | any three of a kind | 13.89% |
| sic bo | 5, 16 | 13.89% |
| sic bo | 4, 17 | 15.28% |
| sic bo | three of a kind | 16.20% |
| sic bo | two-dice combination | 16.67% |
| sic bo | 6, 15 | 16.67% |
| sic bo | two of a kind | 18.52% |
| sic bo | 9, 12 | 18.98% |
| slots | dollar slots (good) | 4.00% |
| slots | quarter slots (good) | 5.00% |
| slots | dollar slots (average) | 6.00% |
| slots | quarter slots (average) | 8.00% |
| sports betting | bet $11 to win $10 | 4.55% |
| three card poker | pair plus | 2.32% |
| three card poker | ante | 3.37% |
| video poker | selected machines | -0.50% |
| *house advantages under typical conditions, expressed "per hand" and including ties, where appropriate. Optimal strategy assumed unless otherwise noted. | ||
Note: this summary is the intellectual property of the author and the university of nevada, las vegas. Do not use or reproduce without proper citation and permission.
Cabot, anthony N., and hannum, robert C. (2002). Gaming regulation and mathematics: A marriage of necessity, john marshall law review, vol. 35, no. 3, pp. 333-358.
Cabot, anthony N. (1996). Casino gaming: policy, economics, and regulation, UNLV international gaming institute, las vegas, NV.
Eadington, william R., and cornelius, judy (eds.) (1999). The business of gaming: economic and management issues, institute for the study of gambling and commercial gaming, university of nevada, reno, NV.
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Bob hannum is a professor of risk analysis & gaming at the university of denver where he teaches courses in probability, statistics, risk, and the theory of gambling. His publications includepractical casino math (co-authored with anthony N. Cabot) and numerous articles in scholarly and gaming industry journals. Hannum regularly speaks on casino mathematics to audiences around the globe. (some of this guide has been excerpted from practical casino math.)
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