1. The output of a manufacturing firm producing metal stampings is 60% top grade and 40%average grade.

a. In drawing a sample of two metal stampings, what is the probability of drawing one top grade and one average grade?

b. In drawing a sample of three metal stampings, what is the probability of drawing two top grade and one average grade?

2. At a parking lot there are 100 vehicles, 60 of which are cars, 30 are vans and the remainder are trucks. If every vehicle is likely to leave what is the probability of.

a. Van leaving first

b. Truck leaving first

c. Car leaving second if either a van or truck left first.

3. Admissions to a local collage university is determined by a national test. The scores on this test are normally distributed with a mean of 500 and a standard deviation of 100. Archie wants to be admitted to the university and he knows he must score better than at least 70% of the students who took the test. Archie take the test and scores 585. Will he be admitted to this university?

4. The length of life of an instrument produced by a machine had a normal distribution with a mean of 12 months and a standard deviation of 2 months. Find the probability that an instrument produced by this machine will last.

a. Less than 7 months

b. Between 7 and 12 months

5. The time taken to assemble a car in a certain plant is a random variable having a normal distribution of 20 hours and a standard deviation of 2 hours. What is the probability that a car can be assembled at this plant in a period of time.

a. Less than 19.5 hours?

b. Between 20 and 22 hours?

6. A bank’s loan officer rates applications for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected find the probability of a rating that is between 150 and 230?

7. Airlines know that only a certain percentage of passengers who have made a reservation on a particular flight will actually take that flight. Consequently, most airlines overbook they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight that the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations. Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive.

Build a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to fins an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company’s revenue is maximized. Consider various alternatives for handling “bumped” passengers, and discuss their effect on the optimal strategy. You will likely determine that different strategies are optimal for different size plans, different trip lengths, etc. Additionally, write a short memorandum to the airline’s CEO summarizing your findings and analysis.

Note: numerous references to the airline overbooking problem can be found online. Read ideas of others and incorporate those that you line and understand in your model. Feel free to stimulate your model using real or synthetic data.