Question

If \( A \) and \( B \) are two non-mutually exclusive events such that \( P(A | B) = P(B | A) \), then:

• A. \( A = B \)
• B. \( A \cap B = \emptyset \)
• C. \( P(A) = P(B) \)
• D. \( A \subseteq B \) but \( A \neq B \)

Hint:

For non-mutually exclusive events, if \( P(A | B) = P(B | A) \), it implies that the probabilities of the two events are the same.

Answer 1

The Correct Option is C

Given that \( P(A | B) = P(B | A) \), we can use the formula for conditional probability: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \quad \text{and} \quad P(B | A) = \frac{P(A \cap B)}{P(A)} \] Setting the two equal to each other: \[ \frac{P(A \cap B)}{P(B)} = \frac{P(A \cap B)}{P(A)} \] This simplifies to: \[ P(A) = P(B) \] Thus, \( P(A) = P(B) \), and the correct answer is option (3).