Question

EYEWITNESS TESTIMONY. A man was struck by a speeding taxi as he crossed the street. An eyewitness testified the taxi (which did not stop) was blue. The man sued the Blue Cab company for his medical expenses. The city has only two cab companies, Blue Cab and Green Cab. Green Cab has 85% of all taxis in the city. At the trial, the man’s lawyer showed that the eyewitness is 80% reliable in identifying the color of taxis. That is, the eyewitness was able to identify the color of a taxi correctly 80% of the time, under conditions similar to those the night of the accident. The lawyer concludes that it is extremely likely that the man was hit by a Blue Cab taxi. Given the eyewitness’s testimony, what is the probability the man was hit by a Blue Cab taxi?

Answer 1

The probability the man was hit by a Blue Cab taxi is 41%.

Explanation:

In terms of bayesian probability, we have to calculate P(B|Wr), or, given the witness saw the right colour, the taxi is from the Blue Cab company.

According to Bayes

P(B|Wr) = P(Wr|B)*P(B)/P(Wr)

P(Wr|B) = 0,8

P(B) = 0.15

To calculate P(Wr), or the probability of the witness of guessing right, we have to consider the two possibilities:

1) The taxi is from Blue Cab (B) and the witness is right (Wr).

2) The taxi is from Green Cab (G) and the witness is wrong (Ww).

The total probality of guessing right is

P(B)*P(Wr) + P(G)*P(Ww) = 0.15*0.8 + 0.85*0.2 = 0.29

So we can calculate:

P(B|Wr) = P(Wr|B)*P(B)/P(Wr) = 0.8*0.15/0.29 = 0.41

The probability the man was hit by a Blue Cab taxi is 41%.