If A and B are two distinct events such that \(P(A|B) = P(B|A)\), then which of the following is/are possible?
(A) \(A = B\)
(B) \(P(A) = P(B)\)
(C) \(A \subset B\) but \(A \ne B\)
(D) \(A \cap B = \phi\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) and (D) only
By the definition of conditional probability, the given condition is
\( \displaystyle \frac{P(A \cap B)}{P(B)} = \frac{P(A \cap B)}{P(A)} \)
\(P(A \cap B)\,[P(A) - P(B)] = 0\)
So the condition holds exactly when \(P(A \cap B) = 0\) or \(P(A) = P(B)\).
Checking the statements
(A) \(A = B\) — Incorrect. A and B are given to be distinct events.
(B) \(P(A) = P(B)\) — Correct. Then both denominators are equal, so \(P(A|B) = P(B|A)\).
(C) \(A \subset B\) but \(A \ne B\) — Incorrect. Then \(P(B|A) = 1\), but \(P(A|B) = \frac{P(A)}{P(B)} \lt 1\) whenever the part of B outside A has non-zero probability.
(D) \(A \cap B = \phi\) — Correct. Then \(P(A \cap B) = 0\), so both conditional probabilities are 0.