Question

Someone witnesses a crime involving a taxi. The witness tells the police that the taxi was blue. The police know that 80% of witnesses are correct and also that 12% of all of the taxis that could have been involved are blue while the rest are green. What is the probability that the taxi was in fact blue? Write your answer out to two decimal places so that 1% is .01, 99% is .99, etc. "base rate fallacy taxi"

Answer 1

Suppose that there's no bias for any particular blue or green taxi to be involved in such incidents. Then consider 100 possible cases (contingencies), in which taxis are involved, in proportion to their numbers. Each case is equally probable. We expect 12 cases to involve blue taxis, 88 to involve green. If a blue taxi were really involved, the witness might report blue (with 80% probability) or green (with 20% probability). If a green taxi were really involved, the witness might report blue (with 20% probability) or green (with 80% probability).

Witness report

Number of cases where Blue taxi Involved and witness report blue = 80% of 12 = 10

Number of cases where Green taxi Involved and witness report blue = 20% of 88 = 18

Number of cases where Blue taxi Involved and witness report green = 20% of 12 = 2

Number of cases where Green taxi Involved and witness report green = 80% of 88 = 70

We have 10+18 = 28 equally likely cases and the probability that a blue taxi really was involved is 10/28 = 0.35