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Every one of us uses the words probable, probability or odds
few times a day in common speech when referring to the possibility of a
certain event happening.
Whether we have math skills or not, we frequently estimate and compare
probabilities, sometimes without realizing it, especially when making
decisions. But probabilities are not just simple numbers attached
objectively or subjectively to events, as they perhaps look, and their
calculus and usage is highly predisposed to qualitative or quantitative
errors in the absence of proper knowledge.
This site is a brief introduction to Probability Theory
Basics and Probability Calculus, with goals of helping nonmathematicians to apply
and perform probability calculus without a teacher and to stimulate them
to go deeper into the notions involved. 









About probability
Beyond
the word probability, which in common speech usually represents a
certain subjective degree of belief in the occurrence of an event, there
is a whole conceptual ensemble developed by Probability Theory.
Moreover, the concept of probability has major philosophical and
psychological implications and several scientific interpretations as
well. But any of its interpretations cannot leave aside the mathematical
notion.
"Call the probability of an event the ratio between the number of
cases that are favorable for that event to occur and the number of all
equally possible cases."  this is the simplistic definition
everyone knows and stands for the definition of probability on a finite
field of events.
Probability Theory extends this definition to a more
complex set of events (sigmafield of events), granted with a certain
mathematical structure, by building a function on this set with specific
properties. This function  called probability  is in fact
a measure on a field of events with values in the interval [0,
1]. Learn more about these notions in the mathematical sections of this
site.


Blaise Pascal
(1623  1662)













James Bernoulli (16541705)
Thomas Bayes (1702  1761)
Pierre Laplace (1749  1827)
Antoine Cournot (1801  1877)
Pafnuty Chebyshev (1821  1894)


Probability Theory
Initially, probability theory was
inspired by games of chance, especially in 17th century France, and was
inaugurated by the Fermat–Pascal correspondence.
However, its complete axiomatization had to wait until Kolmogorov’s Foundations
of the Theory of Probability in 1933. Over time, probability theory found several models in nature and became a
branch of Mathematics with a growing number of applications. In Physics,
probability theory became an important calculus tool at the same time as
Thermodynamics and, later, Quantum Physics.
It has been ascertained that determinist phenomena have a
very small part in surrounding nature. The vast majority of phenomena
from nature and society are stochastic (random).Their study cannot be
deterministic, and that is why hazard science was raised as a necessity.
Probability
theory deals with the laws of evolution of random phenomena. Here are
some examples of random phenomena:
1. The simplest example is the experiment involving
rolling a die; the result of this experiment is the number that appears
on the upper side of the die. Even though we repeat the experiment
several times, we cannot predict which value each roll will take because
it depends on many random elements like the initial impulse of the die,
the die’s position at the start, characteristics of the surface on
which the die is rolled, and so on.
2. A person walks from home to his or her workplace each day.
The time it takes to walk that distance is not constant, but varies
because of random elements (traffic, meteorological conditions, and the
like).
3. We cannot predict the percentage of misfires when firing a
weapon a certain number of times at a target.
4. We cannot
know in advance what numbers will be drawn in a lottery.
In these experiments, the essential conditions of each
experiment are unchanged.
All variations are caused by secondary elements that
influence the result of the experiment. Among the many elements that
occur in the studied phenomena, the theory focuses only on those that
are decisive and ignore the influence of secondary elements. This method
is typical in the study of physical and mechanical phenomena as well as
in technical applications. The randomness and complexity of causes
require special methods of study of random phenomena, and these methods
are elaborated by probability theory.
The application of mathematics in the study of random
phenomena is based on the fact that, by repeating an experiment many
times in identical conditions, the relative frequency of a certain
result (the ratio between number of experiments having one particular
result and total number of experiments) is about the same, and
oscillates around a constant number. If this happens, we can associate a
number with each event; that is, the probability of that event. This
link between structure (the structure of a field of events) and number
is the equivalent of the mathematics of the transfer of quality into
quantity. The problem of converting a field of events into a number is
equivalent to defining a numeric function on this structure, which has
to be a measure of the possibility of an event occurring. Because the
occurrence of an event is probable, this function is named probability.
Probability theory can only be applied to phenomena
that have a certain stability of the relative frequencies around
probability (homogeneous mass phenomena). This is the basis of the
relationship between probability theory and the real world and daily
practice. So, the scientific definition of probability must first
reflect the real evolution of a phenomenon. Probability is not the
expression of the subjective level of man’s trust in the occurrence of
the event, but the objective characterization of the relationship
between conditions and events, or between cause and effect. The
probability of an event makes sense as long as the set of conditions is
left unchanged; any change in these conditions changes the probability
and, consequently, the statistical laws governing the phenomenon.
There are almost no scientific fields in which probability
theory is not applied. Also, sociology uses the calculus of
probabilities as a principal tool. Moreover, some commercial domains are
based on probabilities (insurance, bets, and casinos, among others).


















Probability calculus
To perform probability calculus means to find the numerical
probability of an event, by applying properties of probability and
working the calculations for the specific parameters of the respective
application or problem.
You don't have to be an expert or mathematician for being
able to do probability calculus for finite applications and you don't
have to go deeper in the notions of probability theory. The probability
calculus skills can be developed through algorithmic procedures. The
only things to know priorly are the main definitions and a set of
formulas. Some combinatorial calculus skills are welcome. Besides this
minimal knowledge of probability theory and combinatorics, the only
requirement for the nonmathematician solver is to have a good command
of the four arithmetic operations between real numbers and of basic
algebraic calculus.
Theoretically, any probability calculus problem, no matter
how complex, can be unfolded in successive elementary applications that
use basic formulas, but sometimes finishing the calculus can be very
laborious or even impossible, not to mention the high risk of the
occurrence of errors during a long succession of calculations. The use
of combinatorics and even of classical probability repartitions can
often solve such problems simply and elegantly, whereas the stepbystep
approach is much too laborious and is predisposed to calculation errors.
Every solution of a probability application submits to a
basic algorithm, which basically ensures the correctness of framing and
approach to the calculus problem and of the application of the
theoretical results as well. Even though the methods of solving a
problem can be multiple, all procedures are applied on the basis of this
general algorithm, which is valid for any finite or discrete probability
application. The solution algorithm consists of three main stages:
framing the problem (establishing the probability field attached to an
experiment, textually defining the events to be measured); establishing
the theoretical procedure (choosing the solving method, selecting the
formulas to use); and the calculus
(numerical or combinatorial calculus and the applications of
formulas). As mentioned above, probability
calculus was developed to answer questions on random phenomena,
including in gambling. In what's possibly the most popular recreational activity today,
gambling (especially games like
poker, roulette,
blackjack, slots, or lottery), probability
theory is ubiquitous. Gambling strategy is erected on probability theory, and players wallow
in discussions on this classical subject.
Sources 
Check
our Books
section for good books explaining the probability
concept and its interpretations and applications for nonmathematicians.
Their goals are to help the reader understand what
probability really means, to teach the reader how to rigorously perform
and apply the probability calculus, even without a solid mathematical
background, and to stimulate the reader to go deeper into the notions
involved. You will find there solid A to Z teaching material
about Probability Theory and the practical person can find all the tools
needed to apply and perform probability calculus without a teacher. You may also
browse the sections of this website for the probability theory
basics with applications, including in gambling.
We also provide online gamblingmath
courses and a gamblingmath
counseling service. 



Andrei Markov (1856  1922)
Emile Borel
(1871  1956)
Richard von Mises(1883  1953)
Andrei N. Kolmogorov (1903  1987)


