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# Show that if A, B, and C are mutually independent, then

ISBN: 9780321923271 41

## Solution for problem 6E Chapter 1.4

Probability and Statistical Inference | 9th Edition

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Problem 6E

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.

Step-by-Step Solution:

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

Step 3 of 4

Step 4 of 4

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Show that if A, B, and C are mutually independent, then

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