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axioms. Let the probability of an event
E
be
P(E)
and let
X
be the universal set that
contains all possible elements of concern. The event here is also an element of the
universal set
X
. The three axioms are the following:
Axiom 1:
The probability of an event is always nonnegative, i.e.,
(2.1)
P(E)
!0
Axiom 2:
The probability of the universal set (or the sample space) is 1.0, i.e.,
P(X)
=1
(2.2)
In terms of the individual element of the set, the Axiom 2 can also be expressed as:
(2.3)
where
n
is the number of all elements of the set.
Axiom 3:
If two events
E
1
and
E
2
are mutually exclusive, the probability of their union
is equal to the summation of their probability, i.e.,
(2.4)
It then follows in general for any events
E
1
and
E
2
that
(2.5)
Obviously, for mutually exclusive events,
P
(
E
1
E
2
)=0 (Fig. 2.2), and Equation (2.5)
reduces to Equation (2.4).
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