BTL reader question: How do you get to 2%?

I have started a FAQ page on the BTL book web site here, and I will cross-post the discussion here on the blog. This will give us an opportunity to have a more interactive discussion, if necessary, with additional comments and questions.

Here is the inaugural installment.

On August 13, 2012, this question came in via Twitter:

Well, here is the answer (and thank you for the question, Tia!)

First the problem. At the bottom of page 74 and going on to the top of page 75 I discuss the question posed in a 1978 New England Journal of Medicine paper by Casscells and colleagues to 60 physicians and physicians-in-training at Harvard Medical School. The problem went like this:
 
"If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5 per cent, what is the chance that a person found to have a positive result actually has the disease, assuming that you know nothing about the person's symptoms or signs?"

The question clearly mimics a disease screening situation. The answer is simple yet elusive. Let us assume that 1,000 people are tested. Among them only 1 person has the actual disease. However, given that the false positive rate is 5%, we also know that out of the 1,000 people tested, 50 will have a false positive test. Assuming that the single person with the disease also has a positive test, we can expect 51 people to test positive. But since only 1 out of these 51 people with a positive test has the disease, the answer to the question above is 1/51=2%. This is a pretty shocking realization, given that a large plurality of the Harvard doctors and trainees chose 95% as their answer. 

So, be careful not to let your intuition override the data when making medical decisions!

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