Being able to prove the Total Prob. Theorem and from there the Bayes Theorem has helped me solve problems (granted from a textbook and not real-life ones).
-> In the problem, the sample space comes divided into mutually disjoint portions.
-> Using the definition of probability, we can compute joint distribution from the a-priori distributions.
-> Using the definition of conditional probability, we can flip around the inference direction.
I usually find that all problems (Naive-Bayes spam, cancer) etc. all show this pattern.
