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?
(In reply to Answer
by K Sengupta)
The color of the taxi involved in the crime can either be blue or green.
Let us assume that the color of the taxi involved in the crime is blue. If so, since 85% of the taxi are blue, it follows that 85%*80% = 68% of the testimony would be accurate, while the remaining 85-68 = 17% would be false.
Let us assume that the color of the taxi involved in the crime is green. If so, since 15% of the taxi are green, it follows that 15%*80% = 12% of the testimony would be accurate, while the remaining 15-12=3% would be false.
Thus, albeit accurately or falsely, precisely 71% would say that the taxi was blue. But, we note that only 68% of the testimonial identification is in fact accurate.
Consequently, the required probability is 68/71.