 # Statistics and Probability Questions

Subject Mathematics General Statistics

## Question

1. The problem with the delivery of new systems is that their reliability is uncertain. Unexpected early failure of a system can be very costly. To combat this problem, the concept of burn-in testing was developed. In burn-in testing, the goal is to operate systems for a short period of time before delivery to the customer. This is done in an effort to reveal early system failures due to manufacturing error. If the burn-in time is too short, no failure is revealed, if the burn-in time is too long useful life of the product is used. Thus the reliability of the delivered system is a function of burn-in time. There is a cost associated with burning-in items, a fixed cost and a variable (per unit time) cost for each system being burned-in. There is also a cost if the system is delivered to the customer and it fails before a warrantee period. The determination of optimal burn-in time for a system before delivery is an important question. Develop an influence diagram for this decision problem.

2. Sixty percent of employees in a certain engineering firm have an engineering degree, thirty percent have a business degree, and twenty percent have both, engineering and business degree. If an employee in the engineering firm is randomly chosen, what is the probability that he or she has an engineering degree or has a business degree, but not both.

3. The probability that a surface to air missile hits its intended target is 0.6. If a target is spotted on the radar and 3 missiles are launched simultaneously,
a. what distribution would you use to determine the probability that the target is destroyed?
b. Find the probability that the target is destroyed?

4. A manufacturing process produces items in a continuous manner. 10 manufactured items are placed in a box. If 3 of 10 are defective and if 5 items are randomly selected from the box and tested and not replaced,
a. what distribution would you use to determine the probability that exactly 2 items are defective?
b. Find the probability that exactly 2 items are defective?

5. The average number of new cases of a newly discovered disease is 0.5 per day, if we have 2 drug treatments for the disease left (one treatment will cure one infected individual).
a. what distribution would you use to determine what is the probability that our treatment will last for at least 3 more days?
b. What is the probability that our treatment will last for at least 3 more days?

6. Items are randomly sampled from a production line and tested for quality. They are classified as “good” or “bad”. We can assume that the production quality of each item is independent. The quality control manager claims that the probability that a “bad” item is produced is 0.05. In a sequential sample of 7 items, the 7th item selected was the 3rd defect.
a. What distribution would you use to calculate the probability that, the 7th item selected was the 3rd defect?
b. What is the probability that the 7th item selected was the 3rd defect?

7. In designing an elicitation procedure for the number of accidents at a plant, the analyst inserts the following instruction
It has been noted at similar plants that the occurrence of serious accidents is relatively low, on the order of 0.02 per month, however, due to the rash of recent minor accidents reported in the newspaper, the probability of minor accidents may be much higher. What is your estimate of the number of minor accidents per month?
a. What expert biases might this type of instruction invoke?
b. If the elicitation is from current plant safety supervisors, what additional bias should you be aware of?

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