Question: In the city of Bedford, there are two taxicab companies. Blue and Green. As you may suppose. Blue cabs are blue and Green cabs are green. The Blue company operates 90% of all cabs in the city and the Green company operates the rest. One dark evening, a pedestrian is killed by a hit-and-run taxicab.
There was one witness to the accident. In court, the witness' ability to distinguish cab colors in the dark is questioned, so he is tested under conditions similar to those in which the accident occurred. If he is shown a green cab, he says it is green 80% of the time and blue 20% of the time. If he is shown a blue cab, he says it is blue 80% of the time and green 20% of the time.
The judge believes that the test accurately represents the witness' performance at the time of the accident, so the probabilities he assign to the events of the accident agree with the frequencies reported by the test.
a. Construct the relevance diagram for the judge's state of information.
b. Construct the probability tree representing the judge's state of information. Label all endpoints, supply all branch probabilities, and calculate and label all endpoint probabilities.
c. Flip the tree. Label all endpoints, supply all branch probabilities, and calculate and label all endpoint probabilities.
d. Draw a relevance diagram corresponding to your result in Part (c).
e. If the witness says "The cab involved in the accident was green," what probability should the judge assign to the cab involved in the accident being green?
f. How does the answer to Part (e) compare to the witness' accuracy on the test? Does this result seem surprising? Why or why not?