Please Refer To The Following Figure In Answering The 571435

Please Refer To The Following Figure In Answering The Problemin Th

Please refer to the following figure in answering the 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, then 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.

Paper For Above instruction

The decision-making process in oil exploration, particularly in wildcatting, involves significant uncertainty and risk. Applying Bayesian probability and expected monetary value (EMV) calculations allows decision-makers to evaluate potential sites more objectively and make more informed choices. This paper aims to analyze a common scenario using Bayesian updating, expected value calculations, and decision analysis to determine the optimal site selection based on test results and prior probabilities.

Introduction

Wildcat drilling for oil involves high stakes, where choosing the right site can lead to substantial financial gains or losses. Traditionally, decisions in oil exploration are made based on geological surveys, expert judgment, and probabilistic models. The introduction of probabilistic reasoning, particularly Bayesian methods, enhances the decision process by allowing incorporation of new information—such as core sample analysis results—into prior hypotheses about the presence of oil-bearing structures like domes. This approach improves the probability assessments and subsequent decisions, reducing the risk of dry wells and unnecessary investments.

Bayesian Updating in Oil Wildcatting

In the provided scenario, the company assesses Site 1 using a core sample test that is more reliable in detecting domes than in ruling out their absence. The test’s accuracy and prior probabilities form the basis for Bayesian updating. Given the probabilities P(+ | Dome) = 0.99 and P(– | No Dome) = 0.85, and assuming prior probabilities P(Dome) and P(No Dome), Bayesian methods allow the computation of posterior probabilities P(Dome | +) and P(Dome | –), which reflect the updated belief after test results.

Application of Bayes' Theorem

Bayes' theorem is fundamental in updating prior beliefs in light of new evidence. It states:

\( P(Dome | +) = \frac{P(+ | Dome) \times P(Dome)}{P(+)} \)

where \( P(+) = P(+ | Dome) \times P(Dome) + P(+ | No Dome) \times P(No Dome) \). Similarly, for negative outcomes:

\( P(Dome | -) = \frac{P(- | Dome) \times P(Dome)}{P(-)} \)

These formulas allow the calculation of posterior probabilities once prior probabilities and test accuracy are known. The prior probability \( P(Dome) \) depends on geological data or expert estimates, often assumed to be 0.5 in the absence of other information.

Decision Analysis and Expected Value Calculation

In the context of the oil wildcatting problem, the expected value (EV) of drilling at each site depends on the probability of oil discovery and the associated financial outcomes. When a test indicates the presence of a dome, the expected value is calculated considering both the probability the dome truly exists (posterior probability) and the payoffs for success or failure. If the test is negative, the same approach applies using the updated probabilities.

Calculations involve multiplying the probabilities of each outcome by their respective payoffs, summing the results, and comparing the expected values across different decision options. This quantitative framework guides decision-makers to choose the site with the highest expected monetary value after considering test results and prior knowledge.

Conclusion

The application of Bayesian inference and expected value analysis provides a robust framework for decision-making in oil exploration. It allows the integration of new geological data into initial beliefs, thus refining strategies and minimizing risks. The choice of site following positive or negative test results hinges on the computed posterior probabilities and their impact on the expected financial outcomes. Implementing such probabilistic decision models can significantly enhance the success rate of wildcat drilling endeavors.

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