Solution
The Taxi Cab Problem
Show the problem again
In a city, 85% of cabs are Green and 15% are Blue. A witness to a hit-and-run identifies the cab as Blue, and testing shows the witness correctly identifies cab colors 80% of the time under similar conditions. What is the probability the cab was actually Blue? Express as a fraction.
Worked solution
The answer is 12/29 ≈ 41%. Bayes with 100 cabs: of the 15 Blue, the witness says "Blue" for 12; of the 85 Green, the witness errs on 17, also saying "Blue." So P(Blue | says Blue) = 12/(12 + 17) = 12/29 ≈ 41%: the cab is more likely Green despite the testimony. Kahneman and Tversky's classic demonstration that base rates must temper even fairly reliable evidence.
Source: The taxicab problem, from Daniel Kahneman and Amos Tversky's work on base-rate neglect (1972 onward). Statement written for AxiomIQ.