Correct Answer: A $(P(E^c)P(F^c) \neq P(E^c \cap F^c))$ Here is a step-by-step explanation of why this is the correct choice. Step 1: Understand the Given Condition The problem starts with the condition \( P(E) = P(E|F) \). This is the fundamental definition of two events \(E\) and \(F\) being independent. Using the formula for conditional probability, \[ P(E|F) = \frac{P(E \cap F)}{P(F)} \] If \( P(E) = P(E|F) \), then \[ P(E) = \frac{P(E \cap F)}{P(F)} \] Multiplying both sides by \( P(F) \), \[ P(E)P(F) = P(E \cap F) \] This is the standard condition for independence. So the question becomes: Which of the following is NOT equivalent to \(E\) and \(F\) being independent? Step 2: Evaluate Each Option D. \(E\) and \(F\) are independent This directly matches the derived condition. This statement is equivalent. B. \(P(F) = P(F|E)\) Using conditional probability, \[ P(F|E) = \frac{P(F \cap E)}{P(E)} = \frac{P(F)P(E)}{P(E)} = P(F) \] Thus, this statement is equivalent. C. \(E^c\) and \(F\) are independent If two events \(E\) and \(F\) are independent, then \[ P(E^c \cap F) = P(E^c)P(F) \] which shows \(E^c\) and \(F\) are also independent. This statement is equivalent. A. \(P(E^c)P(F^c) \neq P(E^c \cap F^c)\) If \(E\) and \(F\) are independent, then their complements are also independent: \[ P(E^c)P(F^c) = P(E^c \cap F^c) \] Option A states the opposite inequality, meaning \(E^c\) and \(F^c\) are not independent. This contradicts independence. Conclusion Options B, C, and D are equivalent statements or direct consequences of independence. Option A is not equivalent. Therefore, the correct answer is A.