Problem 3: Was it a mule or a horse? A town has two taxi companies. Blue Mules r

Question

Problem 3: Was it a mule or a horse?

A town has two taxi companies. Blue Mules runs 25 dark blue cabs. Dark Horse operates 75 grey cabs. One dark foggy night, a taxi is involved in a hit-and-run accident. The town's taxis of both companies were all on the streets at the time of the accident. A witness claims to have seen a blue taxi drive away from the scene.

The inspector in charge, Mr Cluesoh remembers Bayes theorem. He makes the witness take a vision test in dark conditions. Presented repeatedly with random blue or grey cars, she correctly identifies grey cars 17 out of 20 times and blue cars 7 out of 10 times.

Mr Cluesoh has now all what’s needed to compute the odds ratio that a Blue Mule did it vs a Dark Horse did it. He turns the data over to you to finish the job.

Use the notation for the events: M: a Blue Mules taxi did it,   H: a Dark Horse taxi did it, SM: Witness sees a blue taxi. SH: Witness sees a grey taxi.

Compute p(M|SM) and p(H|SM), make sure to show all intermediate steps.

Explanation / Answer

P(SM )=P(M)*P(SM|M)+P(H)*P(SM|H) =(25/100)*(7/10)+(75/100)*(1-17/20)=23/80

hence P(M|SM)=P(M)*P(SM|M)/P(SM) =(25/100)*(7/10)/(23/80)=14/23

and P(H|SM) =P(H)*P(SM|H)/P(SM)=(75/100)*(1-17/20)/(23/80)=9/23

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