Environmental Engineering Reference
In-Depth Information
The set is any set of objects of arbitrary nature, each of which is called
an element of the set.
Suppose that some experiments (trial) are carried out and the result
is not known beforehand. Then the set Ω of all possible outcomes of the
experiment is a space of elementary events, and each of its elements α∈Ω
(individual outcome of the experiment) is an
elementary event
. Any set of
elementary events (any combination) is considered a
subset
(part) of the set
Ω and is a
random event
, i.e. any event
A
is a subset of the set Ω:
A
ÌΩ.
In general, if the set Ω contains
n
elements, then it is possible to specify
2
n
subsets (events).
Several definitions will be introduced.
Joint (incompatible) events are such events where the occurrence of one
of them does not preclude (exclude) the possibility of occurrence of another.
Dependent (independent) events - events where the occurrence of one
them affects (does not affect) the occurrence of another event.
The opposite event relative to a selected event
A
is an event which does
not lead to the occurrence of the selected event (indicated by
Ā
).
The complete group of events is a collection of events at which at least
one of the events of this set should occur as a result of experiments.
Axioms of probability theory
. The probability of event
A
is denoted
by
P
(
A
) or
P
{
A
}. The probability is selected so that it meets the following
conditions or axioms:
P
(Ω) = 1;
P
(∅) = 0;
[1.1]
P
(∅) ≤
P
(
A
) ≤
P
(Ω) = 0.
[1.2]
If
A
i
and
A
j
are mutually exclusive events, i.e.
A
i
∧
A
j
= ∅, then
PA A PA P
∨
[1.3]
where
∨
is the sign of logical addition of the events; ∅ is the empty (no
events).
Axiom (3) can be generalised to any number of mutually exclusive
events
{
(
) =
(
) +
(
),
i
j
i
j
n
ii
A
} = 1:
{ }
n
n
=
∑
P A
PA
(
).
[1.4]
i
i
i
=
1
i
=
1
The frequency definition of probability of any event A
:
P(A
)
= m
A
/
n
,
[1.5]
represents the ratio of the number of cases (
m
A
), enabling the occurrence of
an event
A
, to the total number of cases (the number of possible outcomes
of an experiment)
n
.
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