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A blue taxi II (Posted on 2013-03-08) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Solution- independent of n | Comment 2 of 4 |

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 !!!



  Posted by Dan Rosen on 2013-03-13 08:02:48
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