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Question
Let A and B be two events such that P(A) = `3/8`, P(B) = `5/8` and P(A ∪ B) = `3/4`. Then P(A|B).P(A′|B) is equal to ______.
Options
`2/5`
`3/8`
`3/20`
`6/25`
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Solution
Let A and B be two events such that P(A) = `3/8`, P(B) = `5/8` and P(A ∪ B) = `3/4`. Then P(A|B).P(A′|B) is equal to `6/25`.
Explanation:
Given that: P(A) = `3/8`, P(B) = `5/8` and P(A ∪ B) = `3/4`
∴ P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
`3/4 = 3/8 + 5/8 - "P"("A" ∩ "B")`
⇒ P(A ∩ B) = `3/8 + 5/8 - 3/4 = 1/4`
Now `"P"("A"/"B") * "P"("A'"/"B") = ("P"("A" ∩ "B"))/("P"("B")) * ("P"("A'" ∩ "B"))/("P"("B"))`
= `("P"("A" ∩ "B"))/("P"("B")) * ("P"("B") - "P"("A" ∩ "B"))/("P"("B"))`
= `(1/4)/(5/8) * ((5/8 - 1/4))/(5/8)`
= `2/5 * 3/5`
= `6/25`
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