It's really tough to choose one out of so many theorems. However, Bayes' theorem which I learned in my probability class is one of these that dazzles me. I especially like its alternative form:
$ P(F|E) = \frac{P(E | F)\, P(F)}{P(E|F) P(F) + P(E|F^C) P(F^C)}. \! $
Here, E and F are events from sample space S: P(F)!=0, P(E)!=0. P(F|E) is the conditional probability of F given E. P(E), P(F) are marginal probabilities of E and F respectively. F^C is the complementary event of F.
This theorem helped me a lot in programming competitions like TopCoder and I once solved the problem from past Amazon interviews applying it. Click here for more details.
