1. **Problem Statement:** We pick two cards simultaneously from a 52-card deck. Define $X$ as the number of face cards (Jack, Queen, King) in the hand and $Y$ as the number of 10s in the hand. We want to determine if $X$ and $Y$ are dependent or independent. 2. **Key Information:** - Total cards: 52 - Face cards: 12 (4 Jacks + 4 Queens + 4 Kings) - Cards with value 10: 4 - Hand size: 2 cards 3. **Recall the definition of independence:** Two random variables $X$ and $Y$ are independent if and only if for all values $x,y$, $$P(X=x \text{ and } Y=y) = P(X=x) \times P(Y=y)$$ If this equality fails for any pair $(x,y)$, then $X$ and $Y$ are dependent. 4. **Possible values:** - $X$ can be 0, 1, or 2 (number of face cards in 2 cards) - $Y$ can be 0, 1, or 2 (number of 10s in 2 cards) 5. **Calculate marginal probabilities:** - $P(X=0)$: Probability no face cards in 2 cards Number of non-face cards = $52 - 12 = 40$ $$P(X=0) = \frac{\binom{40}{2}}{\binom{52}{2}} = \frac{780}{1326} \approx 0.588$$ - $P(X=1)$: Exactly one face card $$P(X=1) = \frac{\binom{12}{1} \times \binom{40}{1}}{\binom{52}{2}} = \frac{480}{1326} \approx 0.362$$ - $P(X=2)$: Both cards face cards $$P(X=2) = \frac{\binom{12}{2}}{\binom{52}{2}} = \frac{66}{1326} \approx 0.050$$ - $P(Y=0)$: No 10s Number of non-10 cards = $52 - 4 = 48$ $$P(Y=0) = \frac{\binom{48}{2}}{\binom{52}{2}} = \frac{1128}{1326} \approx 0.851$$ - $P(Y=1)$: Exactly one 10 $$P(Y=1) = \frac{\binom{4}{1} \times \binom{48}{1}}{\binom{52}{2}} = \frac{192}{1326} \approx 0.145$$ - $P(Y=2)$: Both cards are 10s $$P(Y=2) = \frac{\binom{4}{2}}{\binom{52}{2}} = \frac{6}{1326} \approx 0.005$$ 6. **Calculate joint probability for a specific pair to test independence:** Check $P(X=1 \text{ and } Y=1)$: probability that the hand has exactly one face card and exactly one 10. Number of ways to pick 1 face card: $\binom{12}{1} = 12$ Number of ways to pick 1 ten: $\binom{4}{1} = 4$ Total ways for this event: $12 \times 4 = 48$ Total ways to pick any 2 cards: $\binom{52}{2} = 1326$ So, $$P(X=1 \text{ and } Y=1) = \frac{48}{1326} \approx 0.0362$$ 7. **Check if $P(X=1 \text{ and } Y=1) = P(X=1) \times P(Y=1)$:** $$P(X=1) \times P(Y=1) = 0.362 \times 0.145 = 0.0525$$ Since $$0.0362 \neq 0.0525,$$ we conclude that $X$ and $Y$ are **not independent**. 8. **Interpretation:** The probability of having exactly one face card and one 10 together is not equal to the product of their individual probabilities, so the number of face cards and number of 10s in the hand are dependent random variables. **Final answer:** $X$ and $Y$ are dependent.