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?
B=The car is blue.
~B=The car is not blue.
X=The person says that it is blue.
~X=The person says that it is not blue.
P(B&X)=0.85(0.8)=0.68
P(B&~X)=0.85(0.2)=0.17
P(~B&X)=0.15(0.2)=0.03
P(~B&~X)=0.15(0.8)=0.12
P(X)=0.68+0.03=0.71
P(BX)=0.68/0.71=68/71=0.9577
Therefore, there is a 95.77% chance that the car is blue, given that the person said that it was blue.
Edited on September 23, 2011, 4:01 pm

Posted by Math Man
on 20110923 16:00:41 