# 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
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%