A crime has occured in Carborough, involving a taxi. The police interviewed an eyewitness, who stated that the taxi involved was blue.

The police know that 85% of taxis in Carborough are blue, the other 15% being green. They also know that statistically witnesses in these situations tend to be correct 80% of the time - which means they report things wrong the other 20% of the time.

What is the probability that the taxi involved in the crime was actually blue?

Before the witness made his comment, there were 4 possible system states:
Taxi was blue and witness saw blue: p = .85 * .8 = .68
Taxi was blue and witness say green: p = .85 * .2 = .17
Taxi was green and witness saw green: p = .15 * .8 = .12
Taxi was green and witness saw blue: p = .15 * .2 = .03

When the witness makes his statement, it eliminates the 2nd and 3rd states above. What is left is that the taxi is blue with p = .68 / .71, and the taxi is green with p = .03 / .71

Previous comments that say that the taxi is blue with p = .8 because the witness is right 80% of the time. This is wrong because when the witness is wrong, is is much more often wrong by saying the taxi is green tnan he is wrong by saying that the taxi is blue.

Posted by Jim Lyon on 2002-05-24 08:00:36