FMM15

Friday Morning Math Problem

Which Color Cab

The psychologists Daniel Kahneman and Amos Tversky used the following example to demonstrate the common failure to evaluate objective probabilities.

A cab is involved in a hit-and-run accident at night. Two cab companies, Blue and Green, operate in a city. You are given the following data:

A) 85% of cabs are green, 15% are blue

B) A witness identified the cab as blue. The reliability of the witness under the given conditions is that they correctly identify the color of the cab 80% of the time.

What is the probability that the cab the witness saw is actually blue?

Solution

This is a classic probability problem where you have to think about a scenario involving two conditions, which means you have to think about $ A \land B $, but also $ \neg A \land \neg B $ We start by calculating two quantities: 1) The probability that the cab's apparent color was correctly identified 2) The probability that the cab's true color was correctly identified Case 1 is the superset, Case 2 is a subset. By taking the ratio of these two, we get the percent of time that the witness, identifying the cab color as blue, is correct because the cab was truly blue. Quantity 1: P(cab's apparent color correctly IDed) = P(cab actually blue) P(blue cab IDed as blue) + P(cab actually green) P(green cab IDed as blue) = (.15)(.80) + (.85)(.20) = (.12) + (.17) = .29 Quantity 2: P(cab's true color correctly IDed) = P(cab actually blue) P(blue cab IDed as blue) = (.15)(.80) = .12 Ratio of these two quantities is the percent of time the ID of a blue cab is because cab is actually blue: Ratio = 41.4%