A taxicab was involved in a fatal hit-and-run accident at night. two cab companies, the green and the blue, operate in the city. you are told that 85% of the cabs in the city are green and 15% are blue and that a witness identified the cab as blue. the court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness was correct in identifying the color of the cab 80% of the time. what is the probability that the cab involved in the incident was blue rather than green?

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Tata

22 November, 00:44

A taxicab was involved in a fatal hit-and-run accident at night. two cab companies, the green and the blue, operate in the city. you are told that 85% of the cabs in the city are green and 15% are blue and that a witness identified the cab as blue. the court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness was correct in identifying the color of the cab 80% of the time. what is the probability that the cab involved in the incident was blue rather than green?

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Myla Cherry · Answer 1

A = Accident caused by blue cab B = witnesses told the cab was blue. It is easy and straightforward to have probability of P (A) = 15% and P (B/A) is 85%. Following the law of total probability. P (D) = P (D|H) P (H) + P (D| ¯H) P (¯H) = 0.8⋅0.15+0.2⋅0.85=0.29 Therefore I get P (H|D) = 41%. Thus, even if the witness said that the cab involved in the accident was Blue, the probability of this being true is only 41%.