Question

Events A and Bare such that P(A) = `1/2`, P(B) = `7/12` and `P(barA ∪ barB) = 1/4`. Find whether the events A and B are independent or not.

Answer 1

Given, P(A) = `1/2`, P(B) = `7/12`

And `P(barA ∪ barB) = 1/4`

For A and B are independent

P(A ∩ B) = P(A).P(B)  ...(i)

Now, `P(barA ∪ barB)` = P(A ∩ B)

⇒ `P(barA ∪ barB)` = 1 – P(A ∩ B)

⇒ P(A ∩ B) = `1 - P(barA ∪ barB)`

⇒ P(A ∩ B) = `1 - 1/4 = 3/4`  ...(ii)

Now, P(A).P(B) = `1/2 xx 7/12 = 7/24`  ...(iii)

Since from equations (ii) and (ill)

P(A ∩ B) ≠ P(A).P(B)

Therefore, events A and B are not independent.