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Chapter 5: Problem 116

Suppose the owner of a salvage company is considering raising a sunken ship. If successful, the venture will yield a net profit of \(\$ 10\) million. Otherwise, the owner will lose \(\$ 4\) million. Let \(p\) denote the probability of success for this venture. Assume the owner is willing to take the risk to go ahead with this project provided the expected net profit is at least \(\$ 500,000\). a. If \(p=.40\), find the expected net profit. Will the owner be willing to take the risk with this probability of success? b. What is the smallest value of \(p\) for which the owner will take the risk to undertake this project?

Short Answer

Expert verified
For a, if \(p = 0.4\), the expected net profit calculation suggests that the owner should not take the risk as the net profit is below $0.5 million. For b, the smallest value of \(p\) for which the owner will take the risk to undertake the project will be determined by solving the equation set up in Step 3.

Step by step solution

01

Calculate expected net profit for given probability

The expected net profit is computed as follows: \[ E = p * X_{success} + (1-p) * X_{failure} \] Given \(p = 0.4\), \(X_{success} = $10\) million and \(X_{failure} = -$4 million, we insert those values into the formula to get the expected net profit: \[ E = 0.4 * 10 + (1 - 0.4) * (-4) \]
02

Decision based on expected net profit

Once we have calculated the expected net profit, we compare it to the minimum acceptable net profit which is $0.5 million. If the expected net profit is equal or larger than $0.5 million, the owner should be willing to take the risk.
03

Calculate the smallest probability for which the owner will undertake the project

We know that the owner is willing to take the risk if the expected net profit is at least $0.5 million. We plug this value into our expected value formula and solve for \(p\): \[ 0.5 = p * 10 + (1 - p) * (-4) \] Solving this equation will provide us with the smallest value for \(p\) at which the owner is willing to undertake the project.

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