If the events A and B are independent, then which of the following statements are true?
(A) $P(A'B)= [1-P(A)] P(B)$
(B) A and B are mutually exclusive
(C) $P(A) = P(B)$
(D) $P(A'B')= [1-P(A)] [1-P(B)]$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (D) only
Given: A and B are independent
(A) $P(A' \cap B) = P(A') P(B) = (1-P(A)) P(B)$ → Correct
(B) Mutually exclusive events satisfy $P(A \cap B) = 0$; for independent events, $P(A \cap B) = P(A)P(B)$ → Not necessarily 0 → Incorrect
(C) No restriction on $P(A)$ and $P(B)$ for independence → Not necessarily equal → Incorrect
(D) $P(A' \cap B') = P(A') P(B') = (1-P(A))(1-P(B))$ → Correct