Page 205 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes
A
You can verify that setting P(A) = provides a probability. This is the case when every result
S
in the sample space is uniformly likely.

!
Caution Do not assume that every outcome is equally likely without good reason.

Beginning from the axioms we can infer different properties. Optimistically, these will consent with
our instinct regarding probability. The proofs of all of these are simple inferences from the axioms.
Proposition 1: If A is an event then P(Ac) = 1 – P(A).

Proof:
Let A be any event. By meaning of the complement Ac and A are disjoint events.
Since they are disjoint we can apply Axiom 3 to get
P(A  Ac) = P(A) + P(Ac).
But (again by definition of the complement) A  Ac = S so

P(S) = P(A) + P(Ac).
By Axiom 2 P(S) = 1 and so
1 = P(A) + P(Ac).

Rearranging this provides the result.

Notes Observe that every line of the proof is defensible by one of the axioms or a definition
(or is a simple manipulation).
We can utilize the results we have proved to infer further ones like the following corollary.

Corollary 1
P() = 0.
Proof.
By definition of complement Sc = . Hence by Proposition 1
P() = 1 “ P(S) = 1 “ 1 = 0

where the second equality accesses Axiom 2.
Corollary 2
If A is an event then P(A) 1.

Proposition 2: If A and B are events and A B then
P(A) P(B).
Proposition 3: If A = {a1, a2, . . . , an} is an event then

n
 
A
P(A) = P ( )    P a i
i 1

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