# Conditional Probabilities

The Caterpillar Corporation depicts the current status for various classifications of equipment used in a Caterpillar manufacturing plant. The company categorizes equipment usage as Low, Medium, or High, depending upon the number of expected manufacturing applications applicable to its use. Equipment is also subject to availability, depending upon its current condition for usage. The plant considers two categories for suitability for use as: In Working Order, and Under Repair. The plant maintains complete status records for all equipment categories in order to schedule production. Currently, records indicate that 10, 18, and 12 pieces of equipment in Low, Moderate, and High usage categories, respectively are available as In Working Order condition. Similarly, 2, 6, and 8 pieces of equipment in Low, Moderate, and High usage categories, respectively are included in the Under Repair condition.

a. Determine the probability that a randomly selected piece of equipment is a high usage item given that it is in working order.

b. Determine the probability that a randomly selected piece of equipment is under repair given that it is a moderate use item.

c. Determine the probability that a randomly selected piece of equipment is in working order. Determine the probability that a randomly selected piece of equipment is in working order given that it is a low usage item. Explain the difference.

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#### Solution Summary

The solution provides step by step method for the calculation of probabilities and conditional probabilities. Formula for the calculation and Interpretations of the results are also included.

Probability / conditional probability

In your opinion, the best outcome would be to have the communications contract awarded to your company but to have the construction contract awarded to a different company. This is not because you are fond of your competition, but instead because if you were to win both, the result would place some strain on your (relatively) small firm's resources. Of course, if the construction contract were the only one you received, it would be good for the firm. You have placed subjective probabilities of 0.60 and 0.45 on the communications and construction contracts respectively, and believe that these events may be considered statistically independent for the purpose of these calculations.

---Find the probability that the result is your best outcome.

---Given that you are awarded at least one contract, find the conditional probability that (in particular) you are awarded the communications contract. This may not be as straightforward as it might appear.

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