Conditional Probability And Independence

Two events are independent precisely when knowledge of one does not change the probability of the other, formally when P(A | B) = P(A), equivalently P(A ∩ B) = P(A)P(B). This independence test is a conceptual favourite at JEE Advanced. Here P(A | B) = 1/2 is given and P(A) = 1/2 as well, so the conditioning event B leaves A's probability unchanged, satisfying the definition of independence exactly. We may cross-check: P(A ∩ B) = P(A | B)P(B) = (1/2)(1/3) = 1/6, while P(A)P(B) = (1/2)(1/3) = 1/6, confirming the product rule. The dependent-because-less option misreads the equality as an inequality. The mutually-exclusive option is wrong since mutual exclusivity would force P(A ∩ B) = 0, contradicting 1/6. The final option invokes an irrelevant comparison of P(A) and P(B), which has nothing to do with independence. The governing criterion is the equality P(A | B) = P(A), and it is essential to distinguish this from mutual exclusivity, which is in fact the opposite extreme of strong dependence. Plausibility check: independence is consistent here because P(A ∩ B) = 1/6 is strictly positive yet exactly equals the product of the marginals, simultaneously ruling out both mutual exclusivity, which would require a zero intersection, and dependence, which would require the conditional and unconditional probabilities to differ.