6. Kahneman [2011, p. 166] gives the following example.

A cab was involved in a hit-and-run accident at night. Two cab companies, Green and Blue, operate in the city. You are given the following data:

• 85% of the cabs in the city are Green and 15% are Blue.

• A witness identified the cab as Blue. The court tested the reliability of the witness in the circumstances that existed on the night of the accident and concluded that the witness correctly identifies each one of the two colors 80% of the time and fails 20% of the time.

What is the probability that the cab involved in the accident was Blue?

(a) Represent this story as a belief network. Explain all variables and conditional probabilities. What is observed, what is the answer?

(b) Suppose there were three independent witnesses, two of which claimed the cab was Blue and one of whom claimed the cab was Green. Show the corresponding belief network. What is the probability the cab was Blue? What if all three claimed the cab was Blue?

(c) Suppose it was found that the two witnesses who claimed the cab was Blue were not independent, but there was a 60% chance they colluded. (What might this mean?) Show the corresponding belief network, and the relevant probabilities. What is the probability that the cab is Blue, (both for the case where all three witnesses claim that cab was Blue and the case where the other witness claimed the cab was Green)?

(d) In a variant of this scenario, Kahneman [2011, p. 167] replaced the first condition with: “The two companies operate the same number of cabs, but Green cabs are involved in 85% of the accidents.” How can this new scenario be represented as a belief network? Your belief network should allow observations about whether there is an accident as well as the color of the taxi. Show examples of inferences in your network.

Make reasonable choices for anything that is not fully specified. Be explicit about any assumptions you make.