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 .
    The experiments generate outcomes (results). An experiment can have more than one outcome, while a test can have only one outcome.
     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 pre-established 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.
     Ω is called the set of outcomes or the sample space. The random events are elements of P (Ω).
     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, 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
A B and is called the union 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 AB and is called the intersection 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 (opposite) of A and is denoted by A.
     If  AB = Φ, meaning A and B cannot occur simultaneously, we say that A and B are incompatible (mutually exclusive) events.
     If   A B = Ω, 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  Ω = AA and event  Φ = AA.
     The first consists of the occurrence of event A or the occurrence of event A, 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 A, which can never happen. This event is called the impossible event.

Let A, B be events from Σ. We say that event A implies event B and write AB, if, when A occurs, B necessarily occurs.
     If we have  AB  and  BA, we say that events A and B are equivalent and write A = B (this reverts to the equality of the sets of tests that correspond to respective events).

     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 eventsΣ, and is denoted by {Ω, Σ}.

We shoud retain:
     - the events are mathematical objects assimilated with sets;
     - the operations between sets also stand 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 non-mathematicians 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.