Online Slots Math - Understanding The Numbers Can Help You Choose The Best Slot Games, slot games mathematics.
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.
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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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