View source for Arman Suleimenov's and Thomas Bayes' Favorite Theorem MA375Fall2008walther - Rhea

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:

<math>P(F|E) = \frac{P(E | F)\, P(F)}{P(E|F) P(F) + P(E|F^C) P(F^C)}.  \!</math>

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 [http://en.wikipedia.org/wiki/Bayes%27_theorem here] for more details.