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Question: In Problem, calculate the probability that the test is positive and a dome structure exists [P(+ and Dome)]. Now calculate the probability of a positive result, a dome structure, and a dry hole [P(+ and Dome and Dry)]. Finally, calculate P(Dome | + and Dry).
Problem: In the oil-wildcatting problem, suppose that the company could collect information from a drilling core sample and analyze it to determine whether a dome structure exists at Site 1. A positive result would indicate the presence of a dome, and a negative result would indicate the absence of a dome. The test is not perfect, however. The test is highly accurate for detecting a dome; if there is a dome, the test shows a positive result 99% of the time. On the other hand, if there is no dome, the probability of a negative result is only 0.85. Thus, P(+ | Dome) = 0.99 and P(- | No Dome) = 0.85. Use these probabilities, the information given in the example, and Bayes' theorem to find the posterior probabilities P(Dome | +) and P(Dome | -). If the test gives a positive result, which site should be selected? Calculate expected values to support your conclusion. If the test result is negative, which site should be chosen? Again, calculate expected values.
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Again referring to Figure, suppose the decision maker has not yet assessed P(Dry) for Site 2 or P(Dome) for Site 1. Let P(Dry) = p and P(Dome) = q.
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