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OFF TOPIC -- A question of logic

Posted By: joe freedman
Date: Saturday, 2 May 2015, at 4:12 a.m.

"Thinking Fast and Slow," an interesting book by the Nobel laureate Daniel Kahneman, contains the following problem: A taxicab is involved in an accident late at night. In the city at issue, 85% of the cabs are green, and 15% are blue. A witness identified the cab as blue. It has been determined that, under the conditions that existed that night, the witness is 80% accurate in distinguishing between the two colors.

Now, assuming the witness is not lying, what is the probability that the cab is blue?

I would have said 80%, because the witness said it was blue and the witness is right 80% of the time. Why should the "base rate," or the percentage of blue and green cabs in the city be relevant? Of course, if the fleet was 100% green, the witness' opinion would have no relevance. But if there's at least 1 blue cab in the fleet, why isn't the witness' 80% accuracy determinative?

Kahneman says the Bayesian answer is 41%. My guess is that I am wrong on this, but can anyone explain?

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