If A and B are independent events, then which of the following is/are true?
(A) \(\bar{A}\) and B are independent events
(B) \(P(A \cap B) = 0\)
(C) \(\bar{A}\) and \(\bar{B}\) are independent events
(D) \(P(A \cap B) = P(A) + P(B)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
For independent events, \(P(A \cap B) = P(A)\,P(B)\). Then:
\(P(\bar{A} \cap B) = P(B) - P(A \cap B)\)
\(= P(B) - P(A)P(B)\)
\(= (1 - P(A))\,P(B) = P(\bar{A})\,P(B)\)
and, using \(P(A \cup B) = P(A) + P(B) - P(A)P(B)\):
\(P(\bar{A} \cap \bar{B}) = 1 - P(A \cup B)\)
\(= 1 - P(A) - P(B) + P(A)P(B)\)
\(= (1 - P(A))(1 - P(B)) = P(\bar{A})\,P(\bar{B})\)
So \(P(A \cap B) = P(A)P(B)\), and the complements \(\bar{A}\), B and \(\bar{A}\), \(\bar{B}\) are also independent pairs.
Checking the statements
(A) \(\bar{A}\) and B are independent events — Correct. \(P(\bar{A} \cap B) = P(\bar{A})P(B)\).
(B) \(P(A \cap B) = 0\) — Incorrect. \(P(A \cap B) = P(A)P(B)\); with \(P(A) = P(B) = \frac{1}{2}\) it is \(\frac{1}{4}\).
(C) \(\bar{A}\) and \(\bar{B}\) are independent events — Correct. \(P(\bar{A} \cap \bar{B}) = P(\bar{A})P(\bar{B})\).
(D) \(P(A \cap B) = P(A) + P(B)\) — Incorrect. With \(P(A) = P(B) = \frac{1}{2}\), \(P(A \cap B) = \frac{1}{4}\) but \(P(A) + P(B) = 1\).
Tip: Do not confuse independent events (\(P(A \cap B) = P(A)P(B)\)) with mutually exclusive events (\(P(A \cap B) = 0\)); statement (B) describes the second.