1. The problem statement, all variables and given/known data Two cards are drawn in succession without replacement from a standard deck. Show that the events A = ”face card on first draw” and B = ”heart on second draw” are independent. Hint: Write A = A1 ∪ A2, where A1 = ”face card and a heart on first draw” and A2 = ”face card and not a heart on first draw.” 2. Relevant equations Two events A and B are independent if P(A ∩ B) = P(A)P(B), law of total probability, conditional probability, Bayes' Law, etc. 3. The attempt at a solution The unconditional probability of A is P(A) = 12/52. A = A1 ∪ A2, A1 ∩ A2 = ∅, and P(A1 ∪ A2) = P(A1) + P(A2). For this case, I considered A as the "new" sample space and used the law of total probability. P(B | A) = P(B | A1)P(A1) + P(B | A2)P(A2). P(B | A) = (12/51)(3/52) + (13/51)(9/52) = 3/52. P(B ∩ A) = P(B | A)P(A) = (3/52)(12/52) = 36/522. Assuming this is correct so far - I just need to find P(B) to determine if they're independent. So, I let A and ~A be disjoint sets and calculate P(B) = P(B | A)P(A) + P(B | ~A)P(~A) with A = A1 ∪ A2 and ~A = ~A1 ∪ ~A2. P(B) = (3/52) + (10/52) = (13/52). P(A)P(B) = (12/52)(13/52) ≠ 36/522.