Ok, Andrew Chin's answer was not exactly what I was asking for, he did make me rethink my approach, and I was able to come up with a general solution that works for my example and others.
d = number of cards dealt
r = number of required outs
u = number of unknown cards remaining
t = number of outs
then the general formula to compute the probability of getting at least the number of required outs when 'd' cards are dealt is (note the expressions in parenthesis are using combination notation, not fractions. the combin function in excel will evaluate that):
For the example case in my question, plugging in the numbers:
When yields 15.058% when evaluated. The top part of the fraction sums up all possible 3 card hands with 2 or 3 spades, and the denominator expression are the number of possible 3 card hands in a 52 card deck.