Question

Let A and B be two events such that `P(overline(A ∪ B)) = 1/6`, P(A ∩ B) = `1/4` and `P(overlineA) = 1/4`, where `overlineA` stands for complement of event A. Then, events A and B are:

विकल्प

• mutually exclusive and independent

• independent

• not independent

• mutually exclusive

Answer 1

Independent

Explanation:

Given that, P(A ∩ B) = `1/4`, `P(overlineA) = 1/4`

and `P(overline(A ∪ B)) = 1/6`

`implies` 1 – P(A ∪ B) = `1/6`  ...[∵ `P(A) + P(overlineA)` = 1]

`implies` 1 – P(A) – P(B) + P(A ∩ B) = `1/6`

`implies` `P(overlineA) - P(B) + 1/4 = 1/6`

`implies` P(B) = `1/4 + 1/4 - 1/6`

`implies` P(B) = `1/3` and P(A) = `3/4`

Now, P(A ∩ B) = `1/4`

= `3/4 xx 1/3`

= P(A) . P(B)

Hence, events A and B are independent events.