Question

8. Prove that (i) if P(A/B) = 1 then P(ABC) = P(BC)
(ii) if P(A/B) = P(A) then P(A/B) = P(Ā)

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Answer 1

(ii) If P(A/B) = P(A), then prove that

P(Ā/B) = P(Ā).

Proof :

Here, B = ĀB + AB

∵ ĀB and AB are mutually exclusive,

P(B) = P(ĀB) + P(AB)

⇒ P(ĀB) = P(B) - P(AB) .....(1)

Now, P(Ā/B)

= P(ĀB)/P(B), where P(B) ≠ 0

= {P(B) - P(AB)}/P(B) [ by (1) ]

= 1 - P(AB)/P(B)

= 1 - P(A/B)

= 1 - P(A) [ ATQ ]

= P(Ā)

⇒ P(Ā/B) = P(Ā)

Hence, proved.