The conditional probability given that the person already had two Kings is what was getting me sidetracked. I am just seeing this notation and still trying to get it to sink in.

I normally say during the problem, OK the denominator of the probability is always the no. of selections being made choosen from the whole sample space. With more practice I will get better at recognizing which questions place conditions on the solution and which are just wasting words in order to confuse me. The denominator in this problem shows me that 4C2 is the choice, 2 Kings, these occupy two of the 3 cards. Then 48C1 =48, fills in the what's implied by the question which is the 3rd card will be a King, right?

This notation is exactly like what my book has and I want to make sure I know what you are saying about the numerator, so what I think you mean that the numerator is the probability of having 2 Kings and the probability of having 3 Kings. So, 4C3 takes care of having 2 Kings.