The classical definition of probability
Let us consider an urn U that contains n balls, from which
m balls are white and n m balls are black. A ball is
drawn at random. We have n elementary events. Let A be the
event the drawn ball is white. This event can occurs in m
tests,
.
Definition: Call the probability of event
A the ratio between the number of situations (tests) favorable
for A to occur and the number of equally possible situations.
Therefore, P = m/n.
This is the classical, dictionary definition of probability. It
can only be used in experiments having equally possible elementary
events.
Probability on a finite field of events
In the case of an arbitrary finite field of events
{Ω, Σ}, a probability on this field is defined as follows:
Definition: Call probability on Σ a
function
that satisfies the following axioms:
(1)
,
for any ;
(2)
;
(3)
,
for any that
.
Axiom (3) can be generalized through recurrence to
any finite number of mutually
exclusive events: If
,
(i,
j = 1,
, n), then
.
Definition: A finite field of events {Ω,
Σ}, structured with a probability P, is called a finite
probability field (finite probability space) and is denoted by {Ω, Σ,
P}.
Probability as a measure
Definition:
Call a σfield of events a field of events {Ω, Σ} that has the
countable additivity property: any countable union of events from Σ is
still an event from Σ (if ,
then ).
The definition corresponds to the definition of
tribe in measure theory.
Definition: Let {Ω, Σ} be a
σfield of events. Call probability on
{Ω, Σ} a numerical positive function P, defined on Σ, which meets
the following conditions:
1)
2)
,
for any countable family of mutually exclusive events
.
The two conditions from the definition of
probability imply the axioms in the definition of measure. Therefore,
the probability is a measure P with P(Ω) = 1, so it
acquires all the properties of a measure.
Probability as a limit
There is an important theorem that illustrates the
way in which probability models the hazard. We enunciate the Law of
Large Numbers, not in its general mathematical form, in order to avoid
having to define more complex concepts, but in an exemplified particular
form, in a way that everyone can understand. The particular enunciation
is the following classic result, known as Bernoullis Theorem:
Theorem: The relative frequency of the occurrence of a
certain event in a sequence of independent experiments performed under
identical conditions converges toward the probability of that event.
The theorem states that if A is an event,
a
sequence of independent experiments,
the
number of occurrences of event A after the first n experiments, then the sequence of nonnegative numbers
is
convergent and its limit is P(A):
The expression
is
called frequency and the expression
is
called relative frequency.
Example:
Consider the classical experiment of tossing a
coin: Let A be the event the coin falls heads up.
Obviously, P(A) = 1/2. Let us say that event A has
the following occurrences:
after the first throw, 0 occurrences, relative
frequency 0/1;
after the first two throws, 0 occurrences,
relative frequency 0/2;
after the first three throws, 1 occurrence,
relative frequency 1/3;
after the first four throws, 1 occurrence,
relative frequency 1/4;
after the first five throws, 2 occurrences,
relative frequency 2/5;
after the
first n throws, occurrences,
frequency .
The Law of Large Numbers says that the sequence
obtained at the right, namely 0, 0, 1/3, 1/4, 2/5,
is convergent
toward 1/2. In other words, while n is growing, the relative
frequency is approximating 1/2 with higher accuracy. The Law of Large Numbers confers on probability a property of
limit.
Of course, the theorem does not provide information
about the terms of the sequence of relative frequencies, but only about
its limit. In other words, we cannot make a precise prediction on the
event at a certain chronological moment, but we can only know the
behavior of the relative frequency of occurrences at infinity.
You should retain:
● Probability is nothing more than a measure; as
length measures distance and area measures surface, probability measures
aleatory events. As a measure, probability is in fact a function with
certain properties, defined on the field of events generated by an
experiment.
● A probability is characterized not only by the
specific function P, but by the entire aggregate the set of
possible outcomes of the experiment the field of generated events
function P, called probability field; probability makes no sense and
cannot be calculated unless we initially rigorously define the
probability field in which we operate. That is, we can say and compute
the probability to be dealt certain cards at blackjack, but saying
the probability of the sun not rising tomorrow makes no sense.
● Probability is not a punctual numerical value;
textually given an event, we cannot calculate its probability without
including it in a more complex field of events. Probability as a number
is in fact a limit, namely the limit of the sequence of relative
frequencies of occurrences of the event to measure, within a sequence of
independent experiments.
Sources 
The
book UNDERSTANDING
AND CALCULATING THE ODDS: Probability Theory Basics and Calculus Guide for
Beginners, with Applications in Games of Chance and Everyday Life is
addressed to nonmathematicians and builds
a clear image of the probability concept by reconstructing its
mathematical definition step by step through its constituent notions,
starting with fundamental notions like sets, functions, convergence
and measure theory basics. You may find it in the Books
section with a free sample.

