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`

Answer 1

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`