State True or False for the statements in the Exercise. If A and B are independent events, then A′ and B′ are also independent.

Question

Philoid · Accepted Answer

TRUE

As A and B are independent

⇒ P(A ∩ B) = P(A)P(B)

As, P(A’ ∩ B’) = P(A ∪ B)’ {using De morgan’s law}

As, P(A ∪ B)’ = 1 – P(A ∪ B)

As we know P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

⇒ P(A ∪ B)’ = 1 –[ P(A) + P(B) – P(A ∩ B)]

⇒ P(A’ ∩ B’) = 1 - P(A) - P(B) + P(A)P(B) {as A & B are independent}

= [1 – P(A) ]– P(B) (1-P(A)]

⇒ P(A’ ∩ B’) = (1 – P(A))(1 – P(B))

= P(A’)P(B’)

hence proved