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:

(B) and (D) only

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 → possible but not necessary

(B) P(A) = P(B) → Correct

(C) A ⊂ B but A ≠ B → possible if P(A) = P(B) and B contains events of measure zero outside A, theoretically possible, but in standard probability, P(A) = P(B) and A ⊂ B implies A = B → Not generally

(D) A ∩ B = ∅ → Then P(A ∩ B) = 0 → Both sides 0 → Equation holds trivially → Possible

Correct statements: B, D