PROBLEM 6E

Show that if A, B, and C are mutually independent, then the following pairs of events are independent: A and (B ∩ C), A and (B ∪ C), A' and (B ∩ C' ). Show also that A', B', and C' are mutually independent.

Answer :

Step 1 of 4 :

Given, A, B and c are mutually independent events, then the following pairs of events are independent.

A and (, A and (, and (.

The claim is to show that , and are mutually independent.

We have to prove that

(a) P(= P(A) P(B) P(C)

(b) P( = P(A) P(BC)

(c) P( = P() P() P()

Step 2 of 4 :

(a)

Let, P(= P(A) + P - P(

We know that, P( = P( - (i)

= P + P- P(

= P(A) + P(B) - P( + P(A) + P(C) - P( - P(A) -

P(B) - P(C) + P( +P( + P(- P(

By simplifying we get,

P( = P(A) + P(- P( - (ii)

Substitute (ii) in (i)

P( = P(A) + P - (P(A) + P(- P()

= P(

= P(A) + P(B) + P(C)

Therefore, A, B and C are mutually independent