Question

Events A and B are such that `P(A) = 1/2, P(B) = 7/12 and P("not A or not B") = 1/4` . State whether A and B are independent?

Answer 1

P(A - not and B - not) = `P (barA cap barB) = P(barA cup barB) = 1 - P (Acup B)`

= 1 − [P(A) + P(B) − P(A ∩ B)]

⇒ `1/4 = 1 - [1/2 + 7/12 - P(A ∩ B)]`

⇒ P(A ∩ B) = `1/4 - 1 + 1/2`

= `(3 - 12 + 6 + 7)/12`

= `4/12`

= `1/3`

and P(A) . P(B) = `1/2 . 7/12`

= `7/24 ≠ P(A ∩ B)`

⇒ P (A) × P (B) ≠ P (A ∩ B)

Events A and B are not independent.