Grinstead-Snell-Probability · Problem 4.1.26

The key insight is to use the conditional probability formula to connect $P(A|B) = P(B|A)$ to the marginal probabilities, then leverage the constraint that $A$ and $B$ cover the entire sample space. Since $P(A \cup B) = 1$, you can express everything in terms of $P(A)$, $P(B)$, and their intersection. The equal conditional probabilities will force a relationship that, combined with the non-empty intersection requirement, pushes $P(A)$ above the halfway mark.

The key insight is to use the conditional probability formula to connect $P(A|B) = P(B|A)$ to the marginal probabilities, then leverage the constraint that $A$ and $B$ cover the entire sample space. Since $P(A \cup B) = 1$, you can express everything in terms of $P(A)$, $P(B)$, and their intersection. The equal conditional probabilities will force a relationship that, combined with the non-empty intersection requirement, pushes $P(A)$ above the halfway mark.