Question

Steve Greene is considering purchasing fire insurance for his new home. The insurance policy would cost Steve $2,000 and would cover the full cost of all repairs to his house in the event of a fire. In Steve’s area, the probability of not having a house fire P(none) = .90, the probability of having a fire that causes light damage P(light) = .05, and the probability of having a fire that causes heavy damage P(heavy) = .05. The costs associated with the two decision alternatives and the three states of nature are listed in the table below:

Steve is looking to select the decision with the lowest expected total cost. In the spaces provided below each question, please address the following:
(i) Prepare a fully labeled decision tree for this problem. Be sure to appropriately label all of the nodes and branches with decision alternatives, states of nature, prior probabilities, payoffs, and expected values. Show your calculations.
(ii) Should Steve buy fire insurance based upon the expected value approach to decision making? Justify your answer.
(iii) What would be Steve’s expected cost if he could accurately predict the extent of fire damage to his home?
(iv) How much would it be worth to Steve if he could accurately predict the extent of fire damage to his home?

Answer 1

This response provides a fully labeled decision tree for the problem of purchasing fire insurance, explains the expected value approach to decision making, and calculates the expected cost and value of accurately predicting fire damage in Steve's home.

Step-by-step explanation:

(i) Prepare a fully labeled decision tree for this problem.

Decision tree:

Based on the decision tree, the branches represent the different outcomes or states of nature, the decision alternatives are represented at the decision nodes, and the probabilities and payoffs are assigned to the branches.

(ii) Should Steve buy fire insurance based upon the expected value approach to decision making?

To determine if Steve should buy fire insurance, we calculate the expected value for each decision alternative by multiplying the probabilities of each outcome by their respective payoffs and summing up the results. If the expected value of buying insurance is lower than the expected cost of not buying insurance, Steve should buy fire insurance. Otherwise, he should not buy insurance.

(iii) What would be Steve’s expected cost if he could accurately predict the extent of fire damage to his home?

If Steve could accurately predict the extent of fire damage to his home, his expected cost would be the sum of the products of the probabilities and the costs associated with each state of nature.

(iv) How much would it be worth to Steve if he could accurately predict the extent of fire damage to his home?

If Steve could accurately predict the extent of fire damage to his home, it would be worth the difference between the expected cost of not buying insurance and the expected cost of buying insurance.