What is Bayes's theorem?

Scientific American explains what the theorem is all about (and does it well), and explores what it can say about the existence of god. This is how it begins:

A patient goes to see a doctor. The doctor performs a test with 99 percent reliability--that is, 99 percent of people who are sick test positive and 99 percent of the healthy people test negative. The doctor knows that only 1 percent of the people in the country are sick. Now the question is: if the patient tests positive, what are the chances the patient is sick?

The intuitive answer is 99 percent, but the correct answer is 50 percent, and Bayes's theorem gives us the relationship between what we know and what we want to know in this problem. ...