1. Problem: Determine which pairs of events are independent when drawing a single card from a standard 52-card deck. 2. Recall that two events A and B are independent if $$P(A \cap B) = P(A) \times P(B)$$. 3. Define events: - Hearts = 13 cards (red), so $$P(Hearts) = \frac{13}{52} = \frac{1}{4}$$ - Black = 26 cards (spades and clubs), so $$P(Black) = \frac{26}{52} = \frac{1}{2}$$ - Ace = 4 cards, so $$P(Ace) = \frac{4}{52} = \frac{1}{13}$$ - Set {2, 3, 9} means drawing a card of rank 2, 3, or 9; each rank has 4 suits, total 12 cards, so $$P(\{2,3,9\}) = \frac{12}{52} = \frac{3}{13}$$ - Red cards = 26 (hearts and diamonds), so $$P(Red) = \frac{26}{52} = \frac{1}{2}$$ 4. Check pairs: (1) Hearts and Black: $$P(Hearts \cap Black) = 0$$ since no card is both hearts and black $$P(Hearts) \times P(Black) = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$$ Since $0 \neq \frac{1}{8}$, they are not independent. (2) Black and Ace: $$P(Black \cap Ace) = P(Black~Ace) = \frac{2}{52} = \frac{1}{26}$$ (because there are 2 black aces: spades and clubs) $$P(Black) \times P(Ace) = \frac{1}{2} \times \frac{1}{13} = \frac{1}{26}$$ Since both probabilities are equal, these events are independent. (3) {2, 3, 9} and Red: The intersection is the cards that are red and of rank 2, 3, or 9. There are 3 ranks and 2 red suits (hearts and diamonds), total $$3 \times 2 = 6$$ cards. $$P(\{2,3,9\} \cap Red) = \frac{6}{52} = \frac{3}{26}$$ $$P(\{2,3,9\}) \times P(Red) = \frac{3}{13} \times \frac{1}{2} = \frac{3}{26}$$ Since both probabilities are equal, these events are independent. 5. Final answer: - Pair (1) is not independent. - Pairs (2) and (3) are independent.