The most commonly given example of Bayes theorem is testing for rare diseases. The results are not intuitive. If a disease is rare, then your probability of having the disease given a positive test result remains low. For example, suppose a disease effects 0.1% of the population and a test for the disease is 95% accurate. Then your probability of having the disease given that you test positive is only about 2%.

Textbooks typically rush through the medical testing example, though I believe it takes a more details and numeric examples for it to sink in. I know I didn’t really get it the first couple times I saw it presented.

I just posted an article that goes over the medical testing example slowly and in detail: Canonical example of Bayes’ theorem in detail. I take what may be rushed through in half a page of a textbook and expand it to six pages, and I use more numbers and graphs than equations. It’s worth going over this example slowly because once you understand it, you’re well on your way to understanding Bayes’ theorem.