# Economics+395+Problems+Fall+2008+-+Answer+Key+1 - Economics...

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Economics 395 Topics in Risk and Uncertainty Homework Problems Answer Key Problem 1 In the town of Springfield, there are two taxi cab companies: Blue Taxi and Green Taxi. All Blue Taxis are blue, and all Green Taxis are green. Blue Taxi is a larger company: 80% of all taxis in Springfield are Blue Taxis. One night, a pedestrian is hit by a car and is killed. The single witness to the incident insists that the car she saw hit the pedestrian was clearly a taxi, and was colored green. She concludes that it must have been a Green Taxi. She estimated that the taxi was traveling at around 30 mph, and viewed it under normal street lights from a distance of about 30 yards. The family of the deceased pedestrian brings a wrongful death suit against the Green Taxi Company, and the eye-witness testifies. The attorney for the defendant produces evidence that eye-witnesses can make mistakes. His studies show that a green colored car is identified as green only around 3/4 of the time, and a blue colored car is identified as blue only around 2/3 of the time. Use an event tree to determine the probability that the car the witness saw and identified as green was indeed green. If this case is t be decided “on the balance of probabilities”, do you think that Green Taxi should be held liable for the death of the pedestrian? Problem 1 Solution On the next page, an event tree depicts the various ways that two distinct sources of uncertainty are resolved. First, nature determines the color of the taxi (blue or green) and then nature determines whether the witness believes the taxi to be blue or green. The second diagram is the “flipped” decision tree. It shows the two sources of uncertainty resolved in the opposite order. First, we discover what color the witness believes the taxi to be; second we determine whether the taxi was truly blue or green. Of particular interest to us is the probability that the taxi is truly green, conditional on the witness believing the taxi to be green. If the witness believes the taxi to be green, it may be the case that the taxi is truly green and the witness’s observation was correct. The probability of this joint event is (1/5)(3/4)=3/20 (it is marked α in figures 2(a) and 2(b)). Alternatively, the taxi may have been blue and the witness was mistaken. This occurs with probability of (4/5)(1/3) = 4/15 (marked β in figures 1(a) and 1(b)). Summing these together gives the probability that the witness believes an observed taxi to be green (marked γ in figure 1(b)).

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G B 4/5 1/5 1/4 3/4 1/3 2/3 Witness believes G Witness believes G Witness believes B Witness believes B 8/15 4/15 (β) 3/20 (α) Figure 1(a): The event tree G B 7/12 5/12 (γ) 9/25 (δ) 16/25 3/35 32/35 Witness believes G Witness believes B B G 8/15 4/15 (β) 1/20 3/20 (α) Figure 1(b): The “flipped” tree 1/20
Then the probability that the taxi was green, conditional on the witness believing it to be green is calculated as (3/20)/(5/12) = 9/25 = 0.36 (marked as δ in Figure 2(b)).

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