Friday puzzle: toenail cancer

Last week you were asked about an ordering of numbers from 0 to 100. The answer is that they were in alphabetical order, starting with “eight,” “eighteen” and ending with “two,” “zero.” Don’t worry if you didn’t get it or it took you a long time. I’ve seen brilliant people grind to a halt on this problem since they immediately look for deep mathematical relationships.

This week’s puzzle is actually one that has some serious implications because it turns out that when it is given to physicians they typically get it completely wrong*

One in a thousand people have toenail cancer.  A screening test for toenail cancer is 95% effective, in that 95% of people with toenail cancer test positive, and 95% of people who don’t have toenail cancer test negative (there’s actually no reason in real life that for these two numbers to be the same).  In a routine screening for toenail cancer you are told you have tested positive. What is the probability that you actually have toenail cancer?

If you’ve not seen a problem like this before, take a guess (you’ll be wrong) before you try and actually work it out mathematically.

Answer next week

* See, for example, Gigerenzer, G. & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: frequency formats. Psychological Review. 102, 684-701

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