I think you shouldn't use the doctor example nor any math toy problem example at first. Instead, discuss some easy-to-understand examples from everyday situations first, and only then present the math behind it. Then discuss some mathy problems. Then go back to more everyday situations.
At least for people who aren't afraid of math, understanding the math is relatively straightforward: understanding how it applies to their lives is the tricky part.
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.