IE546Spring24_hw5_PracticeOnly

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IE546 – Spring 2024 Homework 5 Practice Only [Q1] (8.10) [Q2] (8.25) It is not necessary to have someone else set up a series of bets against you in order for incoherence to take its toll. It is conceivable that one could inadvertently get oneself into a no-win situation through inattention to certain details and the resulting incoherence, as this problem shows. Suppose that an executive of a venture-capital investment firm is trying to decide how to allocate his funds among three different projects, each of which requires a $100,000 investment. The projects are such that one of the three will definitely succeed, but it is not possible for more than one to succeed. Looking at each project as an investment, the anticipated payoff is good, but not wonderful. If a project succeeds, the payoff will be a net gain of $150,000. Of course, if the project fails, he loses all of the money invested in that project. Because he feels as though he knows nothing about whether a project will succeed or fail, he assigns a probability of 0.5 that each project will succeed, and he decides to invest in each project. a According to his assessed probabilities, what is the expected profit for each project? b What are the possible outcomes of the three investments, and how much will he make in each case? c Do you think he invested wisely? Can you explain why he is in such a predicament? [Q3] (8.12 and 8.15) (a) (b) (Next page) It is said that Napoleon assessed probabilities at the Battle of Waterloo in 1815. His hopes for victory depended on keeping the English and Prussian armies separated. Believing that they had not joined forces on the morning of the fateful battle, he indicated his belief that he had a 90% chance of defeating the English; P(Napoleon Wins) = 0.90. When told later that elements of the Prussian force had joined the English, Napoleon re vised his opinion downward on the basis of this information, but his posterior probability was still 60%; P(Napoleon Wins | Prussian and English Join Forces) = 0.60. Suppose Napoleon were using Bayes' theorem to revise his information. To do so, he would have had to make some judgments about P(Prussian and English Join Forces | Napoleon Wins) and P(Prussian and English Join Forces | Napoleon Loses). In particular, he would have had to judge the ratio of these two probabilities. Based on the prior and posterior probabilities given above, what is that ratio? Assess these fractiles for the following uncertain quantities: 0.05 fractile, 0.25 fractile (first quartile), 0.50 (median), 0.75 fractile (third quartile), and 0.95 fractile. Plot your as sessments to create graphs of your subjective CDFs. The official high temperature at O'Hare International Airport tomorrow. Compare the discrete-approximation methods by doing the following: a Use the extended Pearson-Tukey method to create three-point discrete approxima- tions for the continuous distributions assessed in Problem 8.12. Use the approxima- tions to estimate the expected values of the uncertain quantities. b Repeat part a, but construct five-point discrete approximations using bracket medi- ans. Compare your estimated expected values from the two methods.
[Q4] (11.10) [Q5] (12.7) Consider another oil-wildcatting problem. You have mineral rights on a piece of land that you believe may have oil underground. There is only a 10% chance that you will strike oil if you drill, but the payoff is $200,000. It costs $10,000 to drill. The alternative is not to drill at all, in which case your profit is zero. 1. Draw a decision tree to represent your problem. Should you drill? 2. Calculate EVPI. Use either the decision tree or the influence diagram. 3. Before you drill you might consult a geologist who can assess the promise of the piece of land. She can tell you whether your prospects are "good" or "poor." But she is not a perfect predictor. If there is oil, the conditional probability is 0.95 that she will say prospects are good. If there is no oil, the conditional probability is 0.85 that she will say poor. Draw a decision tree that includes the "Consult Geologist" alternative. Be careful to calculate the appropriate probabilities to include in the decision tree. Finally, calculate the EVII for this geologist. If she charges $7000, what should you do? [Q6] (12.6) A decision maker is working on a problem that requires her to study the uncertainty sur rounding the payoff of an investment. There are three possible levels of payoff — $1000, $5000, and $10,000. As a rough approximation, the decision maker believes that each possible payoff is equally likely. But she is not fully comfortable with the assessment that each probability is exactly 1/3, and so would like to conduct a sensitivity analysis. In fact, she believes that each probability could range from 0 to 1/2. a Show how a Monte Carlo simulation could facilitate a sensitivity analysis of the probabilities of the payoffs. b Suppose the decision maker is willing to say that each of the three probabilities could be chosen from a uniform distribution between 0 and 1. Could you incorporate this information into your simulation? If so, how? If not, explain why not, or what addi- tional information you would need. The claim was made in the chapter that information always has positive value. What do you think of this? Can you imagine any situation in which you would prefer not to have some unknown information revealed? a Suppose you have just visited your physician because of a pain in your abdomen. The doctor has indicated some concern and has ordered some tests whose results the two of you are expecting in a few days. A positive test result will suggest that you may have a life-threatening disease, but even if the test is positive, the doctor would want to confirm it with further tests. Would you want the doctor to tell you the outcome of the test? Why or why not? b Suppose you are selling your house. Real-estate transaction laws require that you dis- close what you know about significant structural defects. Although you know of no such defects, a buyer insists on having a qualified engineer inspect the house. Would you want to know the outcome of the inspection? Why or why not?
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