There are n taxi firms in Carborough (n > 1), each one using a different color for its taxis.
A crime has occured in Carborough, involving a taxi. The police interviewed an eyewitness, who stated that the taxi involved was blue.
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. In other words, faced with a blue taxi 20% of witnesses say it is some other color, chosen at random from the remaining n - 1 possible choices. Also, faced with a non-blue taxi, the 80-20 rule is still applicable.
If 85% of the taxis in Carborough are blue, determine the probability (in terms of n) that the taxi involved in the crime was actually blue.
In the general case (when witness evidence is unknown), there are 4 possible situations (B stands for Blue, O stands for Other):-
Witness Taxi prob. of occurence
B B 0.8x0.85 = 0.68
B O 0.2x0.85 = 0.03
O B 0.2x0.85 = 0.17
O O 0.8x0.15 = 0.12
Given the witness has said Blue, our reference group is made out of only the first 2 cases comprising 0.71 of all possibilities, and the probability of the taxi being indeed blue will therefore be :
P(B/B) = 0.68/0.71 = 0.9577
Clearly, the number n of the taxi firms is of no relevance !!!