Bayes' Theorem can equivalently be written using the definition of joint probability: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ This follows directly from the definition of conditional probability: the probability that event $A$ occurs given that $B$ has occurred equals the probability both events occur together, divided by the probability that $B$ occurs. If $A$ and $B$ are independent, then $P(A \cap B) = P(A) \cdot P(B)$, so the equation simplifies to: $P(A \mid B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A)$ This confirms that for independent events, knowing that $B$ occurred gives no information about $A$.