Parameters and variables of the probability modelsWe denote
by
p the number of
distinct symbols 
of
the machine.
If the machine has blank stops, the blank should be considered as a
symbol among them. Parameter
p is specific to the machine.
We
denote by
n the length of a certain payline.
n is
specific to that payline.
Each slot machine belongs to one of the two
types:
Type A – All reels have the same distribution of symbols;
Type
B – The reels have different numbers of stops and each symbol has
different distributions on the stops of the reels.
In case A, denote
by
t the number of stops on each reel and by
the
distribution (number of instances) of symbol
on
each reel (
);
In
case B, denote by
the
number of stops on reel number
j and by
the
distribution of symbol
on
reel number
j (
and
).
Given
a specific symbol
,
the probability of
occurring
on a reel after a spin is
in
case A and
in
case B, where
j is the number of that reel. Probabilities
,
respectively
are
called
basic probabilities in slots.
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 instance, any bar-symbol twice
or any triple of symbols) and 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 B, 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 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, 
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 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:
in
case A and 
in
case B, (1)
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:
(2)
where
is
the number of instances of
,
and so on, 
is
the number of instances of
(
).
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 (
),
then:
(3)
where
and
are
the basic probabilities (the probability of symbol S
occurring on reel number j, respectively k).
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
and
are
independent, while and
,
as well as and
are
non-independent (for 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 machine
In the previous figure, lines
and
,
and
,
and
,
and therefore ,
,
and ,
are non-independent, since within each of the mentioned groups we
have stops of the same reel on different lines. In 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 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
and
in
the last figure. On the contrary, lines
and
not
intersecting each other in the last but one 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 on the reels, not only their
distributions.
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
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so-called PAR sheet of any slot game on the market.
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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:
|
Winning event |
Case A |
Case B |
|
 –
A specific symbol three times
(for example, ( )) |
table |
formula and tables |
|
 –
Any symbol three times (triple) |
|
|
|
 –
A specific symbol exactly twice
(for example, ( any)) |
table |
formula and tables |
|
 –
Any symbol exactly twice (double) |
|
|
|
 –
A specific symbol exactly once
(for example, (  any
any)) |
|
|
|
 –
Any combination of two specific
symbols
(for example, (mix  &  ) , that is ( ) or ( )) |
tables |
formula |
|
 –
Any combination of at least one
of three specific symbols
(for example, (any bar
any bar any bar ), with three bar symbols
like  ,  ,  ) |
formula |
formula |
(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
through
,
operated 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 |
|
|
|
10. A
specific symbol three times or another specific symbol twice |
table |
formula |
|
11. A
specific symbol three times or another specific symbol once |
|
|
|
12. A
specific symbol three times or another specific symbol at
least once |
table |
formula |
|
13.
A specific symbol three times or
any combination of that symbol with another specific symbol |
table |
formula |
|
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:
|
Winning event |
Case A |
Case B |
|
–
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)) |
table |
formula |
|
–
Any symbol exactly three times
(triple) (on a payline of length at least 3) |
|
|
|
–
Any combination of two specific
symbols (on a payline of length at least 3;
for example, (mix & ) , that is ( ) or ( ) or ( ), for a
payline of length 4) |
tables |
formula |
|
–
Any combination of at least one
of three specific symbols (on a payline of length at least 3; for
example, (any bar any bar
any bar any bar), with three bar symbols
like , , , for a
payline of length 4). |
|
|
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
through
,
operated with or):
|
Winning event |
Case A |
Case B |
|
7. A
specific symbol at least three times |
table |
formula |
|
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 |
tables |
formula |
|
10.
A specific symbol four times or
any combination of that symbol with another specific symbol |
tables |
formula |
|
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
Table
