At my college, there's a week before Spring Semester each year in which anyone who wants to can teach a class on any subject, and students go to whatever ones they feel like. I'm thinking about teaching a class on Bayes' Theorem. It would be informal, one to two hours long, and focused mostly on non-obvious applications of it (epistemology, the representativeness heuristic, etc.)
At the moment, I'm thinking about how to design the class, so I'd appreciate any suggestions as to what content I should cover, the best format, clear ways to explain it, cool things related to Bayes' Theorem, good links, and so forth.
It would be cool if you found a way to work in the existence of Cox's theorem -- when I encountered it, I had never thought about why the laws of probability were given as they are, or if there could be a different consistent way to represent and calculate probability besides multiplying numbers together. So it made a big impression on me.
I don't know how to make that part of a more layman-oriented discussion of probability and epistemology, though.
That's a big point for me too. Show that it solves a problem, instead of being a formalism you just happen to use for reasoning under uncertainty for some unknown reason.