# Kahneman on taxis

The BBC podcast More or Less recently ran an interview with Daniel Kahneman, the psychologist who won a Nobel Prize in economics.

He tells two stories to illustrate some point about people’s intuitive reasoning about probabilities. Here’s a rough, slightly abridged transcript of the relevant part:

I will tell you about a city in which two taxi companies operate. One of the companies operates green cars, and the other operates blue cars. 85% of the cars are green, and 15% are blue. There was a hit-and-run accident at night that clearly involved a taxi, and there was a witness who thought that the car was blue. They tested the accuracy of the witness, and they showed that under similar conditions, the witness was accurate 80% of the time. What is your probability that the cab in question was blue, as the witness said, when blue is the minority company?

Here is a slight variation. The two taxi companies have equal numbers of cabs, but 85% of the accidents are due to the green taxis, and 15% are due to the blue taxis. The rest of the story is the same. Now what is the probability that the cab in the accident was blue?

Let’s not bother doing a detailed calculation. Instead, let me ask a qualitative multiple-choice question. Which of the following is true?

1. The probability that the cab is blue is greater in the first scenario than the second.
2. The probability that the cab is blue is greater in the second scenario than the first.
3. The two probabilities are equal.

This pair of stories is supposed to illustrate ways in which people’s intuition fails them. Supposedly, most people’s intuition strongly leads them to one of the incorrect answers above, and they find the correct one quite counterintuitive. Personally, I found the correct answer to be the intuitive one, but that’s probably because I’ve spent too much time thinking about this sort of thing.

I wanted to leave a bit of space before revealing the correct answer, but here it is:

The correct answer is 3 (the probabilities are equal). If I understand him correctly, Kahneman claims that people’s intuition is that the correct answer is 1 — specifically, people think that the probability is 80% in the first scenario and less than 80% in the second.

I’m curious: is there anyone out there whose intuition would have led them to that conclusion?