Question 3: Uncertainty (15 points) The prevalence of a disease among a certain population is 30%. That is, there is a 30 percent chance that a person randomly selected from the population will have the disease. An imperfect test that costs $250 is available to help identify those who have the disease before actual symptoms appear. Those who have the disease have a 80 percent chance of a positive test result; those who do not have the disease have a 5 percent chance of a positive test. Treatment of the disease before the appearance of symptoms costs $3, 000 and inicts additional costs of $300 on those who do not actually have the disease. Treatment of the disease after symptoms have appeared costs $20, 000. The government is considering the following possible strategies with respect to the disease: o 31. Do not test and do not treat early. 0 S2. Do not test and treat early. I S3.Test and treat early if positive and do not treat early if negative. Find the treatment/'tuting strategy that has the lowest expected costs for a member of the population. Note: In doing this exercise, the following notation may be helpful: Let D indicate pres- ence of the disease, ND absence of the disease, Pos a positive test result, Neg a negative test result, T the if treated and N T if not treated. Thus, we have the following information: P (D) = 0.3 which implies: P(ND] = 0.7 P(Pos | D) = 0.8 which implies: P{Neg I D) = 0.2 P(Pos ' ND) = 0.05 which implies: P{Neg | ND} = 0.95 This information allows calculation of some other useful probabilities: P(T), P{D|T) and P(D|NT)