Experiments, events
An experiment is a type of action that
generates events. An example of an experiment is rolling the die. The
performance of an experiment is called test or trial. The
experiments generate outcomes (results). An experiment can have
more than one outcome, while a test can have only one outcome.
Example: The experiment of rolling the die
can have six outcomes (namely the numbers from 1 to 6 inscribed on die’s
sides).
An event is an arbitrary set of
preestablished outcomes of an experiment. An event can happen (occur)
or not, as result of an experiment. If e is the outcome of a test
and A an event related to the respective experiment, we say that
A happens if e belongs to A and A does not
happen if e does not belong to A.
Example: In the roll of a die, the set of
all possible outcomes is {1, 2, 3, 4, 5, 6}. Some events are: A = {1, 3,
5} (uneven number), B = {1, 2, 3, 4} (less than the number 5), C = {2,
4, 6} (even number). If rolling a 3, events A and B occur.
Denote by Ω the set of all possible outcomes of an
experiment and by
P (Ω) the set of all parts of Ω. Set Ω is called the set of
outcomes or the sample space of the experiment. The random events
are elements of P
(Ω).
Operations and relations between events
On the set Σ of the events associated with an
experiment, we can introduce three operations that correspond to the
logical operations or, and, non. Let A and B be
events from Σ. a) A or B is the event that occurs if, and only
if, one of the events A or B occurs. This event is denoted
by and
is called the union or disjunction of events A and
B. b) A and B is the event that occurs if, and only if,
both events A and B occur. This event is denoted by
and
is called the intersection or conjunction of events A
and B. c) non A is the event that occurs if, and only if,
event A does not occur. This event is called the complement
or opposite of A and is denoted by
.
If A and B cannot occur
simultaneously, we say that A and B are incompatible
or mutually exclusive events and denote it by
. If ,
we say that A and B are collectively exhaustive.
In the set Σ of events associated with a certain
experiment, two events with special significance exist, namely, event and
event .
The first consists of the occurrence of event A or the occurrence
of event ,
which obviously always happens; It is natural to call Ω the sure
event. Event ϕ consists of the occurrence of event A
and the occurrence of event ,
which can never happen. This event is called the impossible event.
Let A and B be events from Σ. We say
that event A implies event B and write
,
if, when A occurs, B necessarily occurs. If we have and
,
we say that events A and B are equivalent and write A
= B.
Definition: An event A from is
said to be compound if there exist two events B and C
from ,
such that and
.
Otherwise, the event A is said to be elementary.
Example: In the experiment of rolling the
die: – Event {3, 5} is compound, because
; – Event {1, 2, 4} is compound, because
; – Events {1}, {2}, {3}, {4}, {5}, {6} are
elementary.
Boole algebras
Definition: Call Boole algebra a nonempty set
A, with
operations ,
,
defined
and meeting the following axioms: 1. ;
(commutativity); 2. ;
(associativity); 3. (absorption); 4. ;
(distributivity); 5. ;
(complementarity), for any
A.
Axiom: The set of events associated to an
experiment is a Boole algebra.
Example: The set of parts
P (Ω) of a nonempty set, with union,
intersection and complement (related to Ω) operations, gets a Boole
algebra structure.
Definition: The Boole algebra of the events
associated to an experiment is called the field of events of that
experiment.
So, the field of events is a set of parts of Ω,
structured as an algebra of events Σ, and is denoted by {Ω, Σ}. Therefore, the events can be represented as sets
and the denotations of their operations as from set theory are entirely
justified.
We should retain: ● the events are mathematical objects assimilated
with sets; ● the operations between sets also hold for events; ● the set of events associated to an experiment has
a Boolean structure with respect to the operations and, or
and non; ● any experiment has a sample space and a field of
events attached.
Read the
next definitions.
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.


