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The mathematics of slots Slot Machine Math Slot machine probability


Slot machine probability

Home Gambling Software Articles Books Advertise. Configurations There is a wide variety of the slot machines slot machine probability regard to parametric design and rules. Slot machine probability 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 slot machine probability to a stop of the reel; a window can show one or more spots having a certain shape and arrangement.

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 slot machine probability 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. For the probability calculus in slotsonly a part of the parameters and properties of the entire configuration of a slot slot machine probability 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        or a type of combinations of symbols for slot machine probability, any bar-symbol twice or any triple of symbols maquinitas casino any outcome is a specific combination of stops on that line.

Therefore, the combination of stops should be naturally taken as an elementary event of the probability field. We have  possible combinations of stops in case A and  possible combinations of symbols on a payline of length n across n reels. In case Slot machine probability, we have the same  number of possible combinations of symbols and  possible combinations of stops for that payline of length n.

With regard to the slot machine probability of the events in respect to the ease of the probability computations, we have:. 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,       on a payline of length 3 is a simple event, defined as "two seven and one orange symbols". 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     ,      ,     and so on consider all the symbols of that machine. Any double or two cherries or two oranges or at least one cherry are also complex events of type 1.

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,       on paylines 1, 3, or 5 is a complex event of type 2 expressed through "two seven and one plum symbols". The event      on at least one payline is also a complex event of type 2.

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 2related to several lines. For instance, any triple on paylines 1 or 2 is a complex event of type 3.

At least one slot machine probability 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 slot machine probability, the general formula of the probability of E is:.

For an event E expressed through the number of instances of each symbol on a payline in slot machine probability A, formula 1 is equivalent to:. Formula 2 can be directly applied for winning events defined through the distribution of all symbols on the payline, in case A. These are slot machine probability events.

For more complex events, we must apply the general formula 1which reverts to counting the number of favorable combinations of stops F Eor, for particular situations, apply formula 2 several times and add the results. In case This web page, 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 learn more here 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 Sslot machine probability. 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 principlealong with formulas 1 and slot machine probability and some approximation methods necessary for the ease of computations. When estimating the probability of an slot machine probability related to several lines, some topological properties of slot machine probability group of lines do count; for instance, the independence learn more here 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 mas 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  and  are independent, while  andas well as  and  are link slot machine probability the last two pairs, the lines have a stop in common.

Non-independent lines in a 4 x 5-display of a 5-reel slot slot machine probability. In the previous figure, slot machine probability  and and andand therefore, andare non-independent, since within each of the mentioned groups we have stops of the same reel on different lines. Gaming corp such configuration, there is no group of independent lines, regardless the shape or other properties of the lines.

An immediate consequence of the definition of independent lines is that if two lines intersect each other that is, they share common stopsthey are non-independent, so any group of lines containing them will be slot machine probability. 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  and  in the last figure.

On the contrary, grand victoria casino and resort  and  not intersecting each other in the last but slot machine probability figure are independent. The non-independent lines intersecting or non-intersecting for which there are non-shared stops belonging to the same reels like lines  and  in the last figure are called linked lines.

For events related to linked lines, the probability estimations are only possible if we know the arrangements of the symbols slot machine probability the reels, not only their distributions. All probabilities were worked out under the following assumptions: 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, Slot machine probability, 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 Gamedealing with collecting statistical data from slot players, using the data to estimate the parameters of the slot machine, refining the estimations with the go here collected data and computing the probabilities and other statistical indicators attached to the payout slot machine probability 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. 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, Probabilitiesfor 3-reel, 5-reel, 9-reel, and reel slot machines.

The standard length of a payline is 3. The common winning events on a payline are:. 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  throughoperated with or:. Winning event Case A Case B 8. A specific symbol at least twice table formula and tables 9. A specific symbol at least once     A specific symbol three times or another specific symbol twice table formula A specific symbol three times or another specific symbol once slot machine probability   A specific symbol three times or another specific symbol at least once table formula A specific symbol three times or any combination of that slot machine probability with another specific symbol click at this page formula A specific slot machine probability twice or another specific symbol once     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.

The standard length of a payline slot machine probability 4, but it could also have the see more 3, 6, 7, or 8. The reel 4 x 4-display slot machine could have 8 to 22 slot machine probability of slot machine probability 4, as follows: 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: Winning event Case A Case Slot machine probability  — A specific symbol four times on a payline of length at least  4; for example,     table formula  — Any symbol four times quadruple; on a payline of length at least 4      — A specific symbol exactly three times on a payline of length at least 3; for example,      any.

The table notes the probabilities of the winning events on a payline of length 4. Winning event Case A Case Slot machine probability 7. A specific symbol at least source times table formula 8.

A specific symbol four times or another specific symbol http://desenecopii.info/stringtown-casino.php times     9. A specific symbol four times or another specific symbol at least three times tables formula A specific symbol four times or any combination of that symbol with another slot machine probability symbol tables formula A specific symbol three times or any combination of at least one of three other specific symbols     Winning events on several slot machine probability For the probabilities of these events, we considered only paylines of the regular length 4 in case A.


What are the Odds? - What are Slot Machine Odds? | HowStuffWorks

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