Bayes' Theorem deals with the calculation of posterior probabilities, which isn't always a natural thing to do. We're used to forward-chaining our probabilities (e.g., if we roll a 3 on a die, what's the probability the second roll will give us a total of 8?). Backward-chaining is less intuitive (e.g. if our total on the die was an 8, what's the probability that the first roll was a 3?). Since the rules of probability involve simple addition and multiplication, they work fine in both directions. The thing that makes posterior probability more difficult is that we simply aren't used to thinking about things that way.

Our chapter reading provides an example of a diagnostic test for a rare disease. The resulting confidence in a positive test result is surprisingly low. Discuss why that is so. What is happening in the interaction of the various probabilities that leads to this outcome?