Car Talk: False Positive?

From the fantastic car talk puzzler division

RAY: There’s a rare disease that’s sweeping through your town. Of all the people who are exposed to it, 0.1 percent of the people actually contract the disease. There are no symptoms until the disease actually occurs. However, there’s a diagnostic test that can detect the presence of the disease up to a year before it strikes. You go to your doctor, and he administers the test. It comes out positive. You say, “I’m done for!” Then you get a little bit encouraged. You say, “Wait a minute, doc, is this test 100 percent accurate?” Your doctor responds, “Well, not really. It’s 95 percent accurate.” In other words, 5 percent of the people who take the test will test positive but they don’t really have the disease. Here’s the question: What are the chances that you actually have the disease?

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4 Responses to Car Talk: False Positive?

  1. Peggy says:

    I think there’s not quite enough info — how likely is it that the test correctly detects the disease if you have it? But using either 95% or 100% there, I still get 1.96% chance you have the disease, to 3 sf.

    I love doing this sort of problem with my students, because lots of them think they might be doctors someday.

    • Joshua says:

      Peggy: you are right, but you can get an upper bound (or an estimate of the upper bound, I suppose) by assuming that all infected people who take the test will get a positive result.

  2. Joshua says:

    I love this question so much that for years I would ask a version of this was a question in all job interviews for candidates applying to analytical/technical positions. In part, it was because I wanted to get more people to think carefully about this scenario and the implications for the medical decisions in their own lives.

    My original source for the idea was Gerd Gigerenzer’s book Calculated Risks: How to Know When Numbers Deceive You. I recommend anything and everything that he has written.

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