A positive HIV test
Somebody goes to the doctor, is tested for HIV, and the result comes back positive. The doctor can tell them how often the test is wrong on a person who is not infected — its false positive rate. They want to know something else: whether they are infected.
Those are two different probabilities, and what separates them is how common the infection is among the people being tested. Change the three numbers below and see how far apart they can get.
The same thing, with Bayes
The same thing, in people
everybody imagined, one square each
infected, test positive infected, test negative not infected, test positive not infected, test negative
only the people who tested positive