Basic properties of the probability function
(P1) For any
,
we have
(P2)
(P3) For any
,
(P4) For any
with
,
we have
(P5) For any
,
we have
(P6) If
,
,
then
(P7) For any
,
we have
(P8) For any
,
we have (P9) If
,
then
This property is also called the
inclusionexclusion principle.
(P10) Let
be
events, with .
Then:
The above properties represent formulas
currently used in probability calculus on a finite field of events.
Property (P9) is the main calculus formula for applications in
finite cases.
Properties of probability on a σfield
In
addition, if {Ω, Σ, P} is a σfield, we also have the
following properties: (P11) For any sequence of events
with
(descending),
we have .
For any sequence of events
with
(ascending),
we have .
(P12) For any sequence of events
,
we have .
(P13) If the sequence of events
is
convergent (),
then .
(P14) Generally,
.
The equality holds only if and only if the events are mutually exclusive.
Independent events. Conditional probability
Let us consider the experiment of tossing two
coins and let A – heads on first coin and B –
heads on second coin be two events. The occurrence of event
A and its probability do not depend on the occurrence of
event B, and vice versa. In this case, events A and
B are said to be independent.
Definition: Events A and B
from the probability field {Ω, Σ, P} are said to be
Pindependent if .
Example: In the previous example of the
experiment of tossing two coins we have: P(A and B)
= P(A)
x P(B)
= (1/2) x
(1/2) = 1/4.
Consider an urn containing four white balls and
three black balls. Two people extract one ball each from the urn.
Let A – first person is extracting a white ball and
B – second person is extracting a white ball be two
events. The probability of event B, in the absence of
information about A, is 4/7. If event A has occurred,
the probability of event B is 1/2, so event B depends
on event A. Therefore, these two events are not independent.
It is natural to call the probability of event B conditional
on event A and to denote it by P(B│A).
Definition: Let {Ω, Σ, P} be a
probability σfield and with
.
Call the probability of event A conditional on event B, the
ratio .
Total probability formula. Bayes’s theorem Definition: Call a complete system of
events a finite or countable family of events
,
with for
any ,
and
.
A complete system of events is therefore a partition of
the sample space Ω.
Example: In the experiment of throwing a
die, the system {1, 2, 3}, {4, 5}, {6} is a complete system of
events, while {1, 2, 3}, {3, 4, 5}, {6} is not, as the first two
events are not exclusive.
Theorem (the total probability
formula): Let be
a complete system of events with
.
For any ,
we have .
Bayes’s Formula (the theorem of
hypotheses): Let be
a complete system of events. The probabilities of these events
(hypotheses) are given before performing an experiment. The
experiment produces another event A. Then,
,
for every .
are
called marginal probabilities and
and
are
called conditional probabilities.
Bayes’s formula shows how the probabilities of
the hypotheses change with the occurrence of event
A. Bayes’s theorem is a main result in
probability theory, which relates the conditional and marginal
probability of two aleatory events A and B. In some
interpretations of probability, Bayes’s theorem explains how to
update or revise beliefs in light of new evidence.
Sources 
All
properties of probability, the main results and theorems,
including the random variables and classical probability
distributions, along with suggestive examples and
applications, are all exposed in a comprehensible manner in
the book UNDERSTANDING
AND CALCULATING THE ODDS: Probability Theory Basics and
Calculus Guide for Beginners, with Applications in Games of
Chance and Everyday Life. You may find it in
the Books section with a free
sample.


