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Question
Given P(A) = 0.4 and P(A ∪ B) = 0.7 Find P(B) if P(A/B) = 0.4
Solution
P(A) = 0.4
P(A ∪ B) = 0.7
P(A/B) = 0.4
(i.e.,) `("P"("A" ∩ "B"))/("P"("B"))` = 0.4
⇒ P(A ∩ B) = 0.4 [P(B)] ...........(i)
But we know P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
P(A ∩ B) = P(A) + P(B) – P(A ∪ B)
⇒ P(A ∩ B) = 0.4 + P(B) – 0.7
= P(B) – 0.3 .........(ii)
From (i) and (ii) (Equating R.H.S) we get
0.4 [P(B)] = P(B) – 0.3
0.3 = P(B)(1 – 0.4)
0.6 (P(B)) = 0.3
⇒ P(B) = `0.3/06`
= `3/6`
= 0.5
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