We need to acquire a piece of equipment however we are

Question

We need to acquire a piece of equipment. However, we are uncertain about its reliability. It might need a low, medium or a high number of repairs during its life. The present value ("PV") of the costs of the equipment over its life would be as follows:- If we buy the equipment and the level of repairs are low, medium, or high, the PV of the costs will be $500K, $800K or $1,500K respectively. If we rent the equipment, the PV of the rent will be $1,000K regardless of the repairs needed. This is because the rental company agrees to maintain the equipment for us. We estimate that the probabilities of the level of repairs being low, medium, or high are 10%, 50% and 40% respectively. Before we acquire the equipment, we have the opportunity to run a test on the equipment over a limited period. The test will suggest that the repairs might be low, medium or high. If the number of repairs over the life of the equipment is actually low, then there is a 70% chance that the test will also suggest that they will be low, a 20% chance that the test will suggest they are medium and a 10% chance that they will be high. If the number of repairs over the life of the equipment is actually medium, then there is a 15% chance that the test will suggest that they will be low, an 80% chance that the test will suggest they are medium and a 5% chance that they will be high. If the number of repairs over the life of the equipment is actually high, then there is a 4% chance that the test will suggest that they will be low, a 6% chance that the test will suggest they are medium and a 90% chance that they will be high. The cost of the test will be $50K.

1. Draw a decision tree to help you make a decision whether or not to conduct the pilot test.
2. Use Bayes' theorem to work out revised probabilities given the results of the test.
3. Should we conduct the test? Why?
4. If we conduct a test and it indicates a low repair rate, should we buy or rent the equipment ? Why?
5. If we conduct a test and it indicates a high repair rate, should we buy or rent the equipment ? Why?
6. What is the value of the information given by the test?
7. What is the cost of the test which will make us indifferent between running the test or not?
8. If we do not conduct the test, should we buy or rent the equipment ? Why?