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
Here \(P(A'B)\) means \(P(A' \cap B)\). For independent events, \(P(A \cap B) = P(A)P(B)\). Then
\(P(A' \cap B) = P(B) - P(A \cap B)\)
\(= P(B) - P(A)P(B) = [1 - P(A)]\,P(B)\)
and
\(P(A' \cap B') = 1 - P(A \cup B)\)
\(= 1 - P(A) - P(B) + P(A)P(B)\)
\(= [1 - P(A)]\,[1 - P(B)]\)
So A′ and B, and A′ and B′, are also independent, while \(P(A \cap B) = P(A)P(B)\) places no condition on the sizes of P(A) and P(B).
Checking the statements
(A) \(P(A'B) = [1 - P(A)]\,P(B)\) — Correct. \(P(B) - P(A)P(B)\).
(B) A and B are mutually exclusive — Incorrect. Mutually exclusive needs \(P(A \cap B) = 0\), but \(P(A)P(B) \ne 0\) when both are non-zero.
(C) \(P(A) = P(B)\) — Incorrect. A head on a coin \(\left(\frac{1}{2}\right)\) and a six on a die \(\left(\frac{1}{6}\right)\) are independent.
(D) \(P(A'B') = [1 - P(A)]\,[1 - P(B)]\) — Correct. \(1 - P(A) - P(B) + P(A)P(B)\).