The taxi problem

A witness sees a crime involving a taxi in Carborough. The witness says that the taxi is blue. It is known from previous research that witnesses are correct 80% of the time when making such statements. The police also know that 85% of the taxis in Carborough are blue, the other 15% being green. What is the probability that a blue taxi was involved in the crime?

Solution

Thanks to Steve Kallenborn from Switzerland for correcting the originally published version of this problem.