1. **Problem Statement:** We have two events when drawing cards from a standard deck of 52 cards with replacement: - Event A: Drawing a Heart on the first draw. - Event B: Drawing an Ace on the second draw. We need to determine if these events are independent and find the probability of both events occurring together, $P(A \cap B)$. 2. **Formula for Independence:** Two events A and B are independent if and only if: $$P(A \cap B) = P(A) \times P(B)$$ 3. **Calculate $P(A)$:** There are 13 Hearts in a deck of 52 cards, so: $$P(A) = \frac{13}{52} = \frac{1}{4}$$ 4. **Calculate $P(B)$:** There are 4 Aces in the deck, so: $$P(B) = \frac{4}{52} = \frac{1}{13}$$ 5. **Calculate $P(A \cap B)$:** Since the card is replaced, the draws are independent, so: $$P(A \cap B) = P(A) \times P(B) = \frac{1}{4} \times \frac{1}{13} = \frac{1}{52}$$ 6. **Check independence:** We verify if: $$P(A \cap B) = P(A) \times P(B)$$ Substituting values: $$\frac{1}{52} = \frac{1}{4} \times \frac{1}{13} = \frac{1}{52}$$ This equality holds true, so events A and B are independent. **Final answers:** - The events are independent. - $P(A \cap B) = \frac{1}{52}$.