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.

Bayes' Theorem:

p(blue given reported blue)
  = p(blue and reported blue) / p(reported blue)
  = .85 * .8 / (.85 * .8 + .15 * .2 * 1/(n-1))
  = 17/25 / (17/25 + 3/(100*(n-1)))
  = 17 / (17 + 75/(100*(n-1)))

Posted by Charlie on 2013-03-08 13:21:39