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 Øa. For example, if a represents the proposition:
"our control system contains 0 faults"
then Øa represents the proposition:
"our control system contains at least 1 fault"
Clearly for any event a, the event
a or Øa
is a certain event, and hence from axiom 2:
P(a or Øa) =1
Since the events a and Øa are mutually exclusive It follows from axiom 3 that
1= P(a or Øa) = P(a) + P(Øa)
and hence that
P(a) = 1 - P(Øa)
Sometimes this is called the 4th axiom, but it follows from the others.