While different
people may give P(*a*)
a different value there are nevertheless certain axioms which should
always hold for internal consistency. These are the axioms of
probability theory (which can be *proved*
to be valid when P(*a*)
represents the frequentist approach):

1. P(*a*)
should be a number between 0 and 1

2. If *a*
represents a certain event then P(*a*)=1.

3. If *a*
and *b*
are mutually exclusive events then P(*a*
or *b*)
= P(*a*)+P(*b*)

(mutually
exclusive means that they cannot both be true at the same time; for
example, if *a*
represents the proposition that our control system contains 0 faults,
while *b*
represents the proposition that out control system contains 1 fault)

For any
event *a*
we denote the negation of *a*
at Ø*a*.
For example, if *a*
represents the proposition:

"our control system contains 0 faults"

then Ø*a*
represents the proposition:

"our control system contains at least 1 fault"

Clearly
for any event *a*,
the event

*a*
or Ø*a*

is a certain event, and hence from axiom 2:

P(*a*
or Ø*a*)
=1

Since the
events *a*
and Ø*a*
are mutually exclusive It follows from axiom 3 that

1= P(*a*
or Ø*a*)
= P(*a*)
+ P(Ø*a*)

and hence that

P(a) = 1 - P(Ø*a*)

Sometimes this is called the 4th axiom, but it follows from the others.