I will show two approaches to this problem. The first is what I call "narrowing the sample space."
When you are asked the probability of one event, given another, there is often an intuitive way to approach it. In this case, the given is that a face card is drawn. Suddenly, the sample space is much smaller than the 52 cards in a standard deck.
There are 3 face cards for each suit (J, Q, K), and there are 4 suits. So there are 12 face cards. For our purposes, then, we can ignore all of the other cards in the deck. They are no longer relevant.
So in our probability ratio, the denominator is 12. So for our numerator, we need only consider that, of those 12 face cards, 3 of them are hearts. So the probability of drawing a heart, given that you drew a face card is 3/12 = 1/4.
A second approach is what I might call the formula-based approach or the "textbook approach." The conditional probability formula is:
P(A | B) = P(A ∩ B) / P(B).
So we need to consider two probabilities: the probability of drawing a face card, and the probability of drawing a face card that is a heart.
P(drawing a face card) = 12/52 = 3/13.
P(drawing a face card that is a heart) = 3/52.
Now we divide the second fraction by the first (and I'm going to use the unreduced version of the numerator, for reasons that I believe will be evident soon):
(3/52) / (12/52) = (3/52) · (52/12)
The 52's cancel out (which is why I left the original fraction unreduced), leaving
12/52 = 1/4.
Both methods are perfectly valid, and, while the formula arguable works in more cases, the "sample space narrowing" approach, I believe, shows more of what is going on at a conceptual level and how the changing of a couple of words really can drastically change the meaning of an expression.
Bonus Concept: There is yet another "sample space narrowing" method that gets to the answer even quicker than my first method. You may have observed that our probability ratio reduced a great deal into a curiously simple fraction. There's a good reason for this. It's actually the same probability as drawing a heart from the whole deck! It would be the same if it was the probability of drawing a heart, given that you had drawn an ace, or given that you had drawn a number between 2 and 7, or any other given parameter that involves a kind. And this is for the simple reason that, in a standard deck, there are 4 of each kind, and 1 of those 4 is a heart. There is a lesson in this: don't let the complicated language and notation cause you to forget your simple intuitions.
