If A and B are independent events, then A′ and B′ are also independent.

Question

State True or False for the statement: If A and B are independent events, then A′ and B′ are also independent. Select the correct answer from above options

Answer 1

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