PROBLEM 4E

Prove parts (b) and (c) of Theorem 1.4-1.

Solution 4E

Step1 of 3:

We have the events A and B are independent events.

We need to prove,

a).A1 and B are independent.

b).A1 and B1 are independent.

Step2 of 3:

a).

Let us assume that the events A and B are independent events

Consider,

P(A1 B) = P(A1/ B)P(B) [because P(A/ B) = ]

= [1 - P(A/B)]P(B)

= P(B) - P(A/B)P(B)

= P(B) - P(A B) [because P(A/ B) = ]

= P(B) - [P(A)P(B)] [because we know that A and B are independent]

= P(B)[1 - P(A)]

= P(B) P(A1).

Therefore, P(A1 B)...