Question

If P(A) = `2/5`, P(B) = `3/10` and P(A ∩ B) = `1/5`, then P(A|B).P(B'|A') is equal to ______.

Options

• `5/6`

• `5/7`

• `25/42`

• 1

Answer 1

If P(A) = `2/5`, P(B) = `3/10` and P(A ∩ B) = `1/5`, then P(A|B).P(B'|A') is equal to `25/42`.

Explanation:

Given that: P(A) = `2/5`, P(B) = `3/10` and P(A ∩ B) = `1/5`

P(A') = `1 - 2/5 = 3/5`

P(B') = `1 - 3/10 = 7/10`

And P(A' ∩ B') = 1 – P(A ∪ B)

= 1 – [P(A) + P(B) – P(A ∩ B)]

= `1 - [2/5 + 3/10 - 1/5]`

= `1 - [1/5 + 3/10]`

= `1 - 5/10`

= `1/2`

∴ `"P"("A'"/"B'") = ("P"("A'" ∩ "B'"))/("P"("B'"))`

= `(1/2)/(7/10)`

= `5/7`

And `"P"("B'"/"A'") = ("P"("A'" ∩ "B'"))/("P"("A'"))`

= `(1/2)/(3/5)`

= `5/6`

∴ `"P"("A'"/"B'")*"P"("B'"/"A'") = 5/7 xx 5/6`

= `25/42`