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April 28, 1998

Instructions for the Test: This test will be closed book. However, you will be allowed to use up to 6 pages of your own notes. You will have 50 minutes. For each problem, you should show how you got your answer. The method of the solution will count more than the numerical values. We encourage you to draw diagrams and to state, explicitly and precisely, the definition of your random variables. Good luck!

Practice Test Notes: Below are 9 practice test questions. The nine questions should provide you with an understanding of the types of concepts that may be on the test. The test will consist of four questions.

Suppose that a consumer testing service rates lawn mowers using a simple binary rating scale: 0 or 1, where 1 is the preferred rating. There are three categories being rated: Ease of operation - 0 means difficult to operate, 1 means easy to operate; Cost – 0 means very expensive and 1 means inexpensive; and Ability to be repaired – 0 means difficult to repair and 1 means easy to repair
  1. In how many ways can a lawn mower be rated by this testing service?
  2. The total score for a lawn mower, Y, is the sum of the ratings over three categories. If a group of lawn mowers have equal probability to be rated as a 0 or a 1 in each category, and each category is considered independent, find the probability distribution for Y (i.e., find P(Y=0), P(Y=1), P(Y=2) and P(Y=3)). Show your work.
  3. Find the expected score, E[X], and the standard deviation, s , of the score for the lawn mowers described in part b.
Because not all airline passengers show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is 0.10, and the passengers behave independently.
  1. What is the probability that every passenger who shows up can take the flight?
  2. What is the probability that the flight departs with empty seats?
  3. What is the expected number of passengers who will show up?
  4. What is the standard deviation in the number of passengers who will show up?
For each of the following situations, identify the appropriate probability distribution and parameter values and set up the problem. You do not need to find a final numerical solution. Hint – start by precisely defining the random variable.
  1. Printed circuit cards are placed in a functional test after being populated with semiconductor chips. A lot contains 150 cards, and 30 are selected without replacement for functional testing. If 12 cards are defective in the lot, find the probability that at least one defective card appears in the sample.
  2. The phone lines to an airline system are occupied 50% of the time. Assume that 15 calls are made to the airline independently. Find the probability that exactly 8 calls are occupied.
  3. You are playing basketball with a friend and taking turns shooting free throws. The first one to make 10 free throws buys lunch. If your probability of making a successful free throw is 0.6, what is the probability you make your tenth free throw on your 13th turn?
  4. A particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents a random trial. Find the probability that the first time the light is green is on the fifth morning you approach it.
The edge roughness of slit paper products increases as knife blades wear. Only 1% of products slit with new blades have rough edges, 4% of products slit with blades of average sharpness have rough edges, and 6% of products slit with worn blades exhibit roughness. Currently, 30% of the blades in manufacturing are new, 60% are of average sharpness, and 10% are worn.
  1. Draw a diagram to represent this scenario.
  2. What is the proportion of products that exhibit edge roughness?
  3. If a paper slit has a rough edge, what is the probability that it was caused by a blade of average sharpness?
A pop-quiz is given unannounced in the first 10 minutes of a statistics class. The quiz has 10 multiple choice questions with four answers each. Assume that Brian comes to class completely unprepared, not having had time to study and thus will simply guess at the correct answer to every question.
  1. What is the probability that he will get a perfect score, that is, all correct?
  2. What is the probability that he will get exactly two questions correct?
  3. What is the probability that he will get at least two questions correct?
A particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents an independent trial. Hint: Define the random variable that is appropriate in order to determine the appropriate probability density function.
  1. What is the probability that the light would be green upon approach for an entire week (5 days in a row)?
  2. What is the probability you would need to go five days before getting a day with a green light?
  3. What is the probability that for any given week (5 days), you would get 2 green lights and three red lights.
Three members of a private country club have been nominated for the office of president. The probability that Mr. Adams will be elected is 0.3. The probability that Mr. Brown will be elected is 0.5. The probability that Ms. Cooper will be elected is 0.2. Should Mr. Adams be elected, the probability of an increase in membership fees is 0.8. Should Mr. Brown or Ms. Cooper be elected, the corresponding probabilities for an increase in fees are 0.1 and 0.4.
  1. Draw a diagram that represents captures this situation.
  2. What is the probability that there will be an increase in membership fees?
  3. If it is known that membership fees have increased, what is the probability that Ms. Cooper was elected?
The highway department has published a model for the number of cracks in Interstate 5. The number of cracks in a highway section that are significant enough to require repair is assumed to follow a Poisson distribution, with a mean of one crack per mile.
  1. What is the probability that there are 3 cracks in one mile of highway?
  2. What is the expected number of cracks in 5 miles, and the standard deviation in the number of cracks in 5 miles.
  3. What is the probability that there is at least one crack that requires repair in 5 miles of highway?
  4. You are on a long car trip and decide to pass time by recording some data about the number of cracks in the highway per mile. How would you transform the data in order to compare your findings with those of the highway department’s model?
Your company has recently purchase a new sorting machine that can concurrently sort five streams of letters in its five sorting compartments. However, because your company did not want to spend the extra money, the sorting machine you purchased has an undesirable behavior. If any of the five sorting compartments gets jammed, the entire machine shuts down. If the probability that a sorting component gets jammed during a day is 0.01 and the sorting compartments jam independently, what is the probability that the sorting machine will shut down during any given day?