### What is the expected number of machines

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Question: You are the maintenance engineer for a plant that manufactures consumer electronic goods. You are just about to leave on your vacation for two weeks, and the boss is concerned about certain machines that have been somewhat unreliable, requiring your expertise to keep them running. The boss has asked you how many of these machines you expect to fail while you are out of town, and you have decided to give him your subjective probability distribution. You have made the following assessments:

1. There is a 0.5 chance that none of the machines will fail.

2. There is an approximate 0.15 chance that two or more will fail.

3. There is virtually no chance that four or more will fail. Being impatient with this slow assessment procedure, you decide to try to fit a theoretical distribution.

a. Many operations researchers would use a Poisson distribution in this case. Why might the Poisson be appropriate? Why might it not be appropriate?

b. Find a Poisson distribution that provides a good representation of your assessed beliefs. Give a specific value for the parameter m.

c. Given your answer to b, what is the expected number of machines that will break down during your absence?

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