Question

If A and B are any two events having P(A ∪ B) = `1/2` and P`(barA) = 2/3`, then the probability of `barA ∩ B` is ______.

Options

• `1/2`

• `2/3`

• `1/6`

• `1/3`

Answer 1

If A and B are any two events having P(A ∪ B) = `1/2` and P`(barA) = 2/3`, then the probability of `barA ∩ B` is `1/6`.

Explanation:

We have P(A ∪ B) = `1/2`

⇒ P(A ∪ (B – A)) = `1/2`

⇒ P(A) + P(B – A) = `1/2`  ......(Since A and B – A are mutually exclusive)

⇒ `1 - P(barA) + P(B - A) = 1/2`

⇒ `1 - 2/3 + P(B - A) = 1/2`

⇒ P(B – A) = `1/6`

⇒ `P(barA ∩ B) = 1/6`   ....(Since `barA` ∩ B = B - A)