TITLE: Probability that ace of spades is at bottom of deck IF ace of hearts is NOT at top
QUESTION [6 upvotes]: What is the probability that the ace of spades is at the bottom of a standard deck of 52 cards given that the ace of hearts is not at the top?

I asked my older brother, and he said it should be $\frac{50}{51} \cdot \frac{1}{51}$ because that's $$\mathbb{P}(A\heartsuit \text{ not at top}) \times \mathbb{P}(A\spadesuit \text{ at bottom}),$$ but I'm not sure if I agree. Shouldn't the $\frac{50}{51}$ be $\frac{50}{52}$?
Thanks you!

REPLY [2 votes]: The ace of hearts has 51 positions available (since it's not at the top).
Having placed it somewhere, there are 51 positions available for ace of spades, so
Pr = P(ace of spades not at bottom)*P(ace of diamonds at bottom)
= 50/51 *1/51 = 50/51²