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) $A ⊂ B$ but $A≠ B$
(D) $A∩ 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
Given: $P(A|B) = P(B|A)$
By definition: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, $P(B|A) = \frac{P(A \cap B)}{P(A)}$
So: $\frac{P(A \cap B)}{P(B)} = \frac{P(A \cap B)}{P(A)} \Rightarrow P(A) = P(B)$ (if $P(A \cap B) \neq 0$)
Check options:
(A) $A=B$ → Not possible because the question clearly states that $A$ and $B$ are distinct events.
(B) $P(A)=P(B)$ → Correct.
(C) $A\subset B$ but $A\ne B$ → Not generally possible here.
(D) $A\cap B=\phi$ → Then $P(A\cap B)=0$, so both conditional probabilities become $0$. Hence the condition holds.