 The socalled
PAR sheets (Probability Accounting Reports), exposing the weighting
of the reels, some of the probabilities associated with the winning
combinations, and other statistical indicators, are kept secret by
slots producers. Such secrecy is insubstantial (Barboianu,
2014). Game researchers can obtain them only through
legal intervention. Since 2007, game
researchers obtained the PAR sheets of some slot games through FIPPA
(Freedom of Information and Protection of Privacy Act) in Canada (Harrigan
& Dixon, 2009). Most of these PAR sheets became public after
researchers studied them; others were also available before 2007
through other channels (see Wilson 2004a,
2004b, 2004c, 2004d,
2004e). Still, these are the PAR sheets of a miniscule part of all
slot games on the market.

 Mathematics provides us with some
statistical methods of retrieving the missing data (number of stops
of each reel, number of distinct symbols, and the distribution of
these symbols on the reels the latter is also called the weighting
of the reels), which are essential for the numerical probability
computations in slots.
 The methods briefly described here
(presented in detail in our book The Mathematics of Slots:
Configurations, Combinations, Probabilities; see the
Books section) can be applied in an
organized professional environment in order to estimate and expose
the parametric configuration of any slot game whose PAR sheet is
missing.
 The application of such methods
involves volunteers committed to the collection of data through
longtime observation and recording and a mathematical unit where
these data are processed.

 Statistical methods for
estimating the parameters of the configuration of a slot machine

 In the next sections we use the
same denotations used in section The
configuration of slot machines.

 The raw approximation
 This method is based on the
wellknown result from probability theory called the Bernoulli's
Theorem, which states that in a sequence of independent experiments
performed under identical conditions, the sequence of the relative
frequencies of the occurrence of an event is convergent toward the
probability of that event.
 Applied to slots, that principle
says that if N is the number of spins of a reel with t
stops where we observe as an outcome a specific symbol S that
is placed on c stops and n(N) is the number of
occurrences of S after the N spins, then the sequence
is
convergent toward the probability of occurrence of S, namely
P(S) = c/t.
 The ratio n / N is the
relative frequency of occurrence of S. It follows that for large
values of N, the relative frequency of occurrence of S
approximates the probability of S occurring. The higher N,
the more accurate this approximation. Obviously, the number of spins
N must be large enough for obtaining good approximations of the
ratios ,
and this is the main issue of this method. As theory does not provide us
with tools for choosing N for a given error range, all we have is
the principle the larger N, the better.
 As one can notice, this method of
approximation based on statistical observation is subject to errors
coming from idealizations and various assumptions, and the error ranges
are not even quantifiable. Given these issues, the best way to use this
method is not for individual records, but cumulating progressively the
records coming from several sources and refining the estimations in
correlation with the increase in total number of spins N. This
principle is also common for the odds calculators based on partial
simulations, used for various games.
 Note that the described method
provides us with approximations of the ratios
(the
basic probabilities) for each reel and not the parameters of the
configuration individually ( and
t). However, knowing the basic probabilities is enough for any
probability computation for a slot game.
 A more accurate approximation of the
ratios and
even of and
t individually is still possible through statistical observation,
using a method which can refine the raw estimations obtained through the
previously described method. Such a method is briefly described in what
follows.


Denominatormatch method

Denote by
the
number of occurrences of symbols
to
respectively
after
N
spins of a reel. There is a slight correlation between the recorded
values
for
various large numbers of spins
N.
Based upon this correlation, we can refine the estimation of the ratios
obtained
through the previous method and also find estimations for
and
t,
by recognizing a numerical pattern across some sequences of fractions
representing the ratios between possible values for
and
t.
 The denominatormatch method is based
on the numerical analysis of the fractions
and
on a fivestep algorithm briefly explained below:
 We write each fraction
as
a chain of equal fractions, having numerators from 1 upward and
denominators not necessarily integers, for every i from 1 to p.
Across the p chains of equal fractions obtained, we choose that
of the minimal length (let m be the minimal length). Then, across
the p chains of equal fractions, we extract m sequences of
fractions (one fraction from each equality chain), having the
denominators the nearest to the denominators from the minimal equality
chain respectively. From the m sequences of fractions obtained,
we choose one sequence of p fractions by applying progressively
the following filtering criteria: having denominators as close to each
other as possible, having the highest number of instances of the same
denominator, and the repeating denominator with the largest share being
an integer. As final step, we adjust the numerators of the final
sequence of fractions, as follows: If the sum of the numerators lies
between the minimum and maximum of the denominators, then we take the
numerators as the symbol distribution on the reel ()
and their sum as the number of stops of the reel (t); if their
sum does not lie in that interval, then through addition or subtraction,
we distribute, proportionally with their values, the difference between
their sum and the integer nearest to the mean of the minimal and maximal
denominator, rounding the added/subtracted quantities to integers. For
our resulting estimation, we take the adjusted numerators as the symbol
distribution on the reel (),
and the integer nearest to the mean of the minimal and maximal
denominators as the number of stops of the reel (t).
 This method provides us with the most
probable number of stops t and associated symbol distribution
of
a reel in a certain probability field; the error range of this
approximation is quantifiable in terms of probability (Barboianu, 2013).


 Practical application
 Regarding the practical application of
the methods through statistical observation, it is obviously an arduous
task, since we have to watch and record spins in numbers of thousands.
For online games, software can be developed to help in such an endeavour.
For physical machines, it is far more difficult to watch and note down
thousands of outcomes just for one reel of a machine, not to mention
that the slots operator might not allow this action.
 Infarom is launching the project
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 socalled PAR sheet of any slot game on
the market.
 We are in the phase of looking for
collaborators and funding for this project. After the mathematical unit
will be established, we will have a webpage dedicated to volunteer
registration. Volunteers will be provided with the results of the game
they watched and will have access to all the results obtained for other
slot games.

Contact
us with subject "slots data project" if you want to be part of our
project.
References
Bărboianu, C. (2013). How to estimate the number of
stops and the symbol distribution on a reel. In Infarom (Ed.), The
Mathematics of Slots: Configurations, Combinations, Probabilities
(pp. 4663). Craiova: Infarom.
Barboianu, C.
(2014). Is the secrecy of the parametric configuration of slot machines
rationally justified? The exposure of the mathematical facts of games of
chance as an ethical obligation. Journal of Gambling Issues, in
press.
Harrigan, K.A., & Dixon, M. (2009). PAR
Sheets, probabilities, and slot machine play: Implications for problem
and nonproblem gambling. Journal of Gambling Issues, 23, pp.
81110.
Wilson, J.
(2004a). PAR
excellence: Improve your edge. Slot
Tech Magazine, February
2004, pp.1623.
Wilson, J.
(2004b). PAR
excellance: Part 2. Slot
Tech Magazine, March
2004, pp.1621.
Wilson, J.
(2004c). PAR
excellance: Part 3. Slot
Tech Magazine, April
2004, pp. 2026.
Wilson, J.
(2004d). PAR
excellence  Improving you game  Part IV. Slot
Tech Magazine, May 2004,
pp. 2124.
Wilson, J.
(2004e). PAR excellance: Part V: The end is
here! Slot
Tech Magazine, June 2004,
pp. 2429.
This entry should be cited as:
Barboianu, C.
(2014).
InformativeEthical Contributions of
Mathematics to Problem Gambling.
Retrieved from
http://probability.infarom.ro/ethical.html.
back to
InformativeEthical Contributions page
Sources 
The
content of this section is based on resources from our
Articles and Books
sections, as well as other published research. 



