Alan Turing used it to decode the German Enigma cipher and arguably save the Allies from losing the Second World War; the U.S. Navy used it to search for a missing H-bomb and to locate Soviet subs; RAND Corporation used it to assess the likelihood of a nuclear accident; and Harvard and Chicago researchers used it to verify the authorship of the Federalist Papers.

I haven't seen any explanation of how these kinds of things were done, including calculations. Eliezer's Intuitive Explanation is good, of course, but the examples are very basic. Anything that is notable, even if it's just a published paper, would (I presume) involve data sets and more complex calculations. Does anyone have any good links to complex examples where they actually go through the math and make it easy to follow?

(I would like to understand this better; plus my father, a molecular biologist, asked me to explain Bayes' Theorem and how to use it to him.)