Goats and a Cadillac

The BBC runs, for the umpteenth time, a story about the Monty Hall Problem, with which I am sure you are all familiar.

It ends with this:

Next year we will need something different, perhaps Simpson's Paradox. Imagine that 1% of people have a certain disease.

A diagnostic test has been developed which performs as follows - if you have the disease, the test has a 99% chance of giving the result "positive", while if you do not have the disease, the test has 2% chance of (falsely) giving the result "positive".

A randomly chosen person takes the test. If they get the result "positive", what is the probability that they actually have the disease? The answer, 1/3, is perhaps surprisingly low.

Answers on a postcard please.

Simple.

If you take a person at random, there is a 99% chance he does not have the disease, but will test will show a positive in 2% of cases anyway.

There is a 1% chance he does have the disease, and the test will be correct 99% of the time.

So if we test 100 people at random, there will be 1.98* false positives and 0.99** true positives, so two-thirds of those 2.97 positives are false.

* 100 people x 99% disease-free x 2% false positives = 1.98
** 100 people x 1 with the disease x 99% true positives = 0.99