**Revealing slots
secrets: Generating a PAR sheet through statistical methods**

by Catalin Barboianu

published in the May
2014 issue of *Slot Tech Magazine*

For decades, the slots game has remained one of the most popular games of chance, despite a specific element that could limit its appeal, namely non-transparency: Players do not know the parametric configurations of the machines they play at, as this information is rarely exposed. Card players know the composition of the decks in play, roulette players know the numbers on the wheel, lottery players know the numbers from which the winning line is drawn, and so on. Slots remains the only game in which players are not aware of the essential parameters of the game, such as number of stops of the reels, number of symbols and their distribution on the reels, which makes the slot games unique in this respect.

Obviously, the lack of data regarding the parametric configuration of a machine prevents people from computing the odds of winning and other mathematical indicators, since the probability formulas hold as variables those parameters. The PAR sheets (Probability Accounting Reports), exposing these parameters of the machines and probabilities associated with the winning combinations are not publicly exposed and can be only retrieved upon request from the game producers, usually via legal intervention (for example, through FIPPA, Freedom of Information and Protection of Privacy Act).

Regarding the possible reasons of the game
producers for keeping the PAR sheets far away from the public, there is a stated
reason expressed by producers who declined the
requests, in their appeals to the court decisions, through the fact
that PAR sheets contain information considered
to be trade secrets in the gaming industry and consist of mathematical formulas
and equations developed by their engineers and that information
significantly prejudices their competitive position. There are debates on
whether such reason is justified and the opponents argue that it
fails against the generality of the math formulas and
equations – although the parametric details vary from game to game, the
mathematical results concerning probability, expected value, and other
statistical indicators are just *applications* of general formulas that are
publicly available in mathematics and common across all slot machines, and no
individual or corporate body can claim ownership of such a pattern or formula;
they also claim that the competitive prejudice reason fails against the open
possibility for all slot producers to configure, test, and use any parametric
design for their slot machines and the producer can manipulate the game
parameters, including the payout schedule, in unlimited ways, so as to obtain
the desired statistical indicators for the house. Another possible reason is the
hypothetical fear of losing players who face the real odds and expected values
of their games, which is criticisable through the *a priori* expectation of
the players for low and very low odds of winning induced by the experienced
secrecy of PAR sheets and through the lottery example, in which lottery players
continue to play against the (well known) lowest odds of winning from all games
of chance due to other addictive elements that slots also hold.

Ongoing studies debate on the ethical aspects of the exposure of the mathematical facts behind games of chance and on whether the exposure should be limited to the parametric configuration, basic numerical results (such as probabilities for basic winning events and expected value) or more advanced mathematical results and their interpretations.

** **Mathematics
has its role in this issue and its main contribution is not only another
argument on the insubstantiality of the secrecy of slots PAR sheets, but a
practical one: Mathematics provides players and professionals with some
statistical methods for retrieving these missing data. Having these data along
with the mathematical formulas, anyone can generate the PAR sheet of any slot
machine.

The configuration of a reel refers to the
distribution of the symbols over the stops of that reel. Denoting by *t*
the number of stops and by *p* the number of distinct symbols
on
the reel, and denoting by the
number of symbol on
the reel (),
then the vector is
called the *distribution of the symbols*
on
the reel, also known as the *weighting* of a reel. Each reel has its own
distribution of symbols. We can assume the same number of distinct symbols on
each reel (*p*) through a convention: if a symbol does not appear on a
reel, we could simply take its distribution on that reel as being zero. A blank
is considered as a distinct symbol within the mathematical model.

** The raw approximation**. This method is
based on the well-known 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. Note
that the described method provides us with approximations of the ratios
(usually
labeled as "hit frequency" in PAR sheets) 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 further.

**Denominator-match 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 denominator-match method is based on the
numerical analysis of the fractions and
on a five-step algorithm, which I describe very briefly in this article and is
explained at large in my last book *The Mathematics of Slots: Configurations,
Combinations, Probabilities*:

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, through an approximation algorithm that leads to one denominator for
all the fractions, being the sum of the numerators, as it is the relation
between *t* and numbers *.*

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.

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.

** **Any information acquired on *t*
besides the presented statistical methods of estimation is useful with respect
to the accuracy of the approximations because it can give a clue as to how high
we should choose *N* for avoiding irrelevant results (for example, if *t*
= 100, we intuit that choosing *N* = 1,000 or lower would not be high
enough for relevant results). Besides the methods based on statistical
observation, there exists a method of estimating *t* through physical
measurements, applicable to some particular types of slot machines. This method
exploits the information given by the appearance of the reel on the display. As
we know, only a small part of the reel (either physical or virtual) is visible
on the display and this part can be seen as one or several adjacent stops
(usually 3, up to 5). So we can view from 1 up to 5 consecutive stops of the
reel. If the appearance of this part of the reel is three-dimensional (which is
possible for both physical and virtual reels), by measuring some parameters of
this image, we can deduce an estimation for the number of stops of that reel (*t*).
Basically, the apparent lengths of the visible stops give full information on
the curvature of the reel, which then leads to an estimation of the entire
number of stops, since the number of visible stops per the circular length of
the visible reel is proportional to the total number of stops per the circular
length of the entire reel. This method can be applied only to reels showing at
least two consecutive stops on the display in three-dimensional view. The method
cannot be applied to virtual reels showing several consecutive stops in flat
image. As in the case of the previous method through statistical observation,
there are issues with the practical application of the method through physical
measurements. There might be technical issues regarding acquiring the proper
position for measurement or placing the measurement tool on the surface of the
machine. Also for this method, an alternative would be for the observer to take
photos and make the measurements on the photos. Of course, the slot machine
operator might not allow the direct measurement and/or taking photos.

An applied mathematics unit using the denominator-match method applied on statistical records for retrieving the missing parametric configurations of the existent slot machines and generating their attached probability results is in project.

With regard to problem gambling, past empirical
studies found that *facing the odds* does not change much gamblers'
behavior toward a decrease, however a clear conclusion is not drawn yet. As
regard to the slots game itself, the odds and other mathematical facts do count
as information in a trivial strategic sense: It is as if someone asks you to bet
you can jump from a high place and land on your feet; of course, it is an
advantage for you to know in advance the height from which you will jump or
measure it before you bet, as you might decline the bet or propose another one
for a certain measurement, and this means *decision*.

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