In a deck of 52 cards, you draw two cards consecutively without replacement. Are the events "drawing a heart on the first draw" and "drawing a black card on the second draw" conditionally independent given that the first card drawn is a heart?
I solved this problem as follows and I would like to know if the approach is correct:
$A: \text{Drawing a heart on the first draw} \\ B: \text{Drawing a black card on the second draw}\\ C: \text{First card drawn is a heart} \\ P(A): \frac{13}{52} = 0.25\\ P(B): \frac{26}{51} \approx 0.5098 \\$
If events $A$ and $B$ are independent then: $P(A\cap B) = P(A)P(B)$.
$P(A\cap B) = \frac{_{13}C_1 \times _{26}C_1}{_{52}C_2} \approx 0.2549\\ P(A)P(B) \approx 0.1275$
Therefore these events are not independent because $P(A)P(B) \neq P(A\cap B)$. Now, we check for conditional independence on $C$.
$P(A|C) = 1$: Because if we know the first card drawn is a heart then it changes the probability of A. Probability of drawing a heart on the first draw given the first draw is a heart is 1.
$P(B|C) = \frac{P(C|B)P(B)}{P(C)}\\ P(C) = P(C|B)P(B) + P(C|B^c)P(B^c) = P(C\cap B) + P(C \cap B^c) = \frac{_{13}C_1 \times _{26}C_1}{_{52}C_2} + \frac{_{13}C_1 \times _{25}C_1}{_{52}C_2} \approx 0.2549 + 0.2451 \approx 0.50\\ P(B|C) = \frac{0.2549}{0.50} \approx 0.5098 \\ P(A \cap B | C) = P(A|C)P(B|C) = 1 \times 0.5098 = 0.5098\text{ must hold for conditional independence.}\\ P(A \cap B|C) = \frac{P(C|A \cap B)P(A \cap B)}{P(C)} = \frac{P(A \cap B \cap C)}{P(C)} = P(A \cap B) = 0.2549.$
Therefore, these events are not independent and, not conditionally independent since, $P(A \cap B|C) \neq P(A|C)P(B|C).$
Is this approach correct. Are there gaps in my understanding that jump out to you? Is there a better way to solve this question?