The configuration of slot machines

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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.

Configuration of the display
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.

 

Horizontal, vertical, oblique, and broken lines

 

Triangular line

Trapezoidal line

 

Zigzag lines

 

Double-stair line

 

Symmetric upper-lower transversal triangular line

The paylines of slot machines can be of any complexity for each slot machine in part, but usually they have particular shapes, following properties such as non-self crossing, colinearity, symmetry, up-down or left-right direction, and/or crossing over the reels.

With respect to probability calculus, there are only the length of a line and the number of particular lines in a given rectangular grid (for complex events involving several paylines) that count, regardless of other properties of them. For instance, the number of horizontal lines included in an m x n-size rectangular grid (m rows and n columns) is given in the next table.

Table of values for the number of horizontal lines in a rectangular grid 

      n

m

2

3

4

5

6

7

8

9

2

0

2

6

12

20

30

42

56

3

0

3

9

18

30

45

63

84

4

0

4

12

24

40

60

84

112

5

0

5

15

30

50

75

105

140

6

0

6

18

36

60

90

126

168

7

0

7

21

42

70

105

147

196

8

0

8

24

48

80

120

168

224

9

0

9

27

54

90

135

189

252

 The number of transversal oblique lines included in an m x n-size rectangular grid is given in the next table.

Table of values for the number of transversal oblique lines in a rectangular grid 

      n

m

2

3

4

5

6

7

8

9

2

0

0

0

0

0

0

0

0

3

0

2

4

6

8

10

12

14

4

0

4

2

4

6

8

10

12

5

0

6

4

2

4

6

8

10

6

0

8

6

4

2

4

6

8

7

0

10

8

6

4

2

4

6

8

0

12

10

8

6

4

2

4

9

0

14

12

10

8

6

4

2

The number of left-right transversal trapezoidal lines included in an m x n-size rectangular grid is given in the next table.

Table of values for the total number of left-right transversal trapezoidal lines in a rectangular grid 

      n

m

2

3

4

5

6

7

8

9

2

0

0

2

2

2

2

2

2

3

0

0

4

4

6

6

6

6

4

0

0

6

6

10

10

12

12

5

0

0

8

8

14

14

18

18

6

0

0

10

10

18

18

24

24

7

0

0

12

12

22

22

30

30

8

0

0

14

14

26

26

36

36

9

0

0

16

16

30

30

42

42

Configuration of the reels

Distribution of the symbols
Each symbol appears a number of times on the reel. Together, they fill all stops of the reel that is, the number of stops is the sum of the numbers of instances each distinct symbol appears on the reel (if the reel has blanks, this statement also holds, as we can consider the blank as a new symbol).
Denote by t the number of stops and by p the number of distinct symbols  on the reel. Denote by  the number of symbols , by  the number of symbols , and so on, by  the number of symbols . Obviously,  and .

Definition: Call the vector   the distribution of the symbols  on the reel.

The distribution of the symbols tells us how many of each symbol is on the reel. Each reel has its own distribution of symbols.

There are  possible distributions of symbols on a t-stops reel, for all possible values of p.

 

 

 

 

Common symbols on the reels

Definition: Let  be a distribution of the p symbols on the t stops of a reel. Call arrangement of the symbols on the reel any function a from the set of stops to the set of distinct symbols, such that  for any i from 1 to p (that is, the number of stops having assigned symbol  by function a is ).
The number of possible arrangements of the symbols on a reel in a given distribution is .

The arrangement of symbols on the reel does not count toward probability estimations, except for events related to paylines that contain stops of the same reel or complex events related to such non-independent paylines. In other words, for estimating odds, usually we do not need to know how the symbols are arranged on the reel, but how many instances of each symbol we have on the reel and their total number (the number of stops).

There are two types of slot machines with respect to the symbol distributions on the reels:

A All reels have the same distribution of symbols; each symbol S has the same distribution on the t stops of each reel, denoted by ;
B The reels have different numbers of stops  and each symbol S has different distributions on the stops of the n reels, denoted by:  on reel 1,  on reel 2, ...,  on reel n.
All probability models and computations should be worked out assuming one of the two cases A and B.

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 Sources

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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