# Problems

[Page 506 ( continued )]
1.

Indicate which of the following probabilities are objective and which are subjective . (Note that in some cases, the probabilities may not be entirely one or the other.)

1. The probability of snow tomorrow

2. The probability of catching a fish

3. The probability of the prime interest rate rising in the coming year

4. The probability that the Cincinnati Reds will win the World Series

5. The probability that demand for a product will be a specific amount next month

6. The probability that a political candidate will win an election

7. The probability that a machine will break down

8. The probability of being dealt four aces in a poker hand

2.

A gambler in Las Vegas is cutting a deck of cards for \$1,000. What is the probability that the card for the gambler will be the following?

1. A face card

2. A queen

3.

Downhill Ski Resort in Colorado has accumulated information from records of the past 30 winters regarding the measurable snowfall. This information is as follows :

Snowfall (in.)

Frequency

019

2

2029

7

3039

8

4049

8

50+

5

30

1. Determine the probability of each event in this frequency distribution.

2. Are all the events in this distribution mutually exclusive? Explain.

4.

Employees in the textile industry can be segmented as follows:

 Employees Number Female and union 12,000 Female and nonunion 25,000 Male and union 21,000 Male and nonunion 42,000

1. [Page 507]
2. Determine the probability of each event in this distribution.

3. Are the events in this distribution mutually exclusive? Explain.

4. What is the probability that an employee is male?

5. Is this experiment collectively exhaustive? Explain.

5.

The quality control process at a manufacturing plant requires that each lot of finished units be sampled for defective items. Twenty units from each lot are inspected. If five or more defective units are found, the lot is rejected. If a lot is known to contain 10% defective items, what is the probability that the lot will be rejected? accepted?

6.

A manufacturing company has 10 machines in continuous operation during a workday . The probability that an individual machine will break down during the day is .10. Determine the probability that during any given day 3 machines will break down.

7.

A polling firm is taking a survey regarding a proposed new law. Of the voters polled, 30% are in favor of the law. If 10 people are surveyed, what is the probability that 4 will indicate that they are opposed to the passage of the new law?

8.

An automobile manufacturer has discovered that 20% of all the transmissions it installed in a particular style of truck one year are defective. It has contacted the owners of these vehicles and asked them to return their trucks to the dealer to check the transmission. The Friendly Auto Mart sold seven of these trucks and has two of the new transmissions in stock. What is the probability that the auto dealer will need to order more new transmissions?

9.

A new county hospital is attempting to determine whether it needs to add a particular specialist to its staff. Five percent of the general hospital population in the county contracts the illness the specialist would treat. If 12 patients check into the hospital in a day, what is the probability that 4 or more will have the illness?

10.

A large research hospital has accumulated statistical data on its patients for an extended period. Researchers have determined that patients who are smokers have an 18% chance of contracting a serious illness such as heart disease, cancer, or emphysema, whereas there is only a .06 probability that a nonsmoker will contract a serious illness. From hospital records, the researchers know that 23% of all hospital patients are smokers, while 77% are nonsmokers. For planning purposes, the hospital physician staff would like to know the probability that a given patient is a smoker if the patient has a serious illness.

11.

Two law firms in a community handle all the cases dealing with consumer suits against companies in the area. The Abercrombie firm takes 40% of all suits , and the Olson firm handles the other 60%. The Abercrombie firm wins 70% of its cases, and the Olson firm wins 60% of its cases.

1. Develop a probability tree showing all marginal, conditional, and joint probabilities.

2. Develop a joint probability table.

3. Using Bayes's rule, determine the probability that the Olson firm handled a particular case, given that the case was won.

12.

The Senate consists of 100 senators , of whom 34 are Republicans and 66 are Democrats. A bill to increase defense appropriations is before the Senate. Thirty-five percent of the Democrats and 70% of the Republicans favor the bill. The bill needs a simple majority to pass. Using a probability tree, determine the probability that the bill will pass.

13.

A retail outlet receives radios from three electrical appliance companies. The outlet receives 20% of its radios from A, 40% from B, and 40% from C. The probability of receiving a defective radio from A is .01; from B, .02; and from C, .08.

1. Develop a probability tree showing all marginal, conditional, and joint probabilities.

2. Develop a joint probability table.

[Page 508]
3. What is the probability that a defective radio returned to the retail store came from company B?

14.

A metropolitan school system consists of three districtsnorth, south, and central. The north district contains 25% of all students, the south district contains 40%, and the central district contains 35%. A minimum-competency test was given to all students; 10% of the north district students failed, 15% of the south district students failed, and 5% of the central district students failed.

1. Develop a probability tree showing all marginal, conditional, and joint probabilities.

2. Develop a joint probability table.

3. What is the probability that a student selected at random failed the test?

15.

A service station owner sells Goodroad tires, which are ordered from a local tire distributor. The distributor receives tires from two plants, A and B. When the owner of the service station receives an order from the distributor, there is a .50 probability that the order consists of tires from plant A or plant B. However, the distributor will not tell the owner which plant the tires come from. The owner knows that 20% of all tires produced at plant A are defective, whereas only 10% of the tires produced at plant B are defective. When an order arrives at the station, the owner is allowed to inspect it briefly . The owner takes this opportunity to inspect one tire to see if it is defective. If the owner believes the tire came from plant A, the order will be sent back. Using Bayes's rule, determine the posterior probability that a tire is from plant A, given that the owner finds that it is defective.

16.

A metropolitan school system consists of two districts, east and west. The east district contains 35% of all students, and the west district contains the other 65%. A vocational aptitude test was given to all students; 10% of the east district students failed, and 25% of the west district students failed. Given that a student failed the test, what is the posterior probability that the student came from the east district?

17.

The Ramshead Pub sells a large quantity of beer every Saturday. From past sales records, the pub has determined the following probabilities for sales:

Barrels

Probability

6

.10

7

.20

8

.40

9

.25

10

.05

1.00

Compute the expected number of barrels that will be sold on Saturday.

18.

The following probabilities for grades in management science have been determined based on past records:

Probability

A

.10

B

.30

C

.40

D

.10

F

.10

1.00

The grades are assigned on a 4.0 scale, where an A is a 4.0, a B a 3.0, and so on. Determine the expected grade and variance for the course.

19.

A market in Boston orders oranges from Florida. The oranges are shipped to Boston from Florida by either railroad , truck, or airplane; an order can take 1, 2, 3, or 4 days to arrive in Boston once it is placed. The following probabilities have been assigned to the number of days it takes to receive an order once it is placed (referred to as lead time) :

[Page 509]

Probability

1

.20

2

.50

3

.20

4

.10

1.00

Compute the expected number of days it takes to receive an order and the standard deviation.

20.

An investment firm is considering two alternative investments, A and B, under two possible future sets of economic conditions, good and poor. There is a .60 probability of good economic conditions occurring and a .40 probability of poor economic conditions occurring. The expected gains and losses under each economic type of conditions are shown in the following table:

Economic Conditions

Investment

Good

Poor

A

\$900,000

\$800,000

B

120,000

70,000

Using the expected value of each investment alternative, determine which should be selected.

21.

An investor is considering two investments, an office building and bonds . The possible returns from each investment and their probabilities are as follows:

Office Building

Bonds

Return

Probability

Return

Probability

\$50,000

.30

\$30,000

.60

60,000

.20

40,000

.40

80,000

.10

1.00

10,000

.30

.10

1.00

Using expected value and standard deviation as a basis for comparison, discuss which of the two investments should be selected.

22.

The Jefferson High School Band Booster Club has organized a raffle. The prize is a \$6,000 car. Two thousand tickets to the raffle are to be sold at \$1 apiece. If a person purchases four tickets, what will be the expected value of the tickets?

23.

The time interval between machine breakdowns in a manufacturing firm is defined according to the following probability distribution:

Time Interval (hr.)

Probability

1

.15

2

.20

3

.40

4

.25

1.00

[Page 510]

Determine the cumulative probability distribution and compute the expected time between machine breakdowns.

24.

The life of an electronic transistor is normally distributed, with a mean of 500 hours and a standard deviation of 80 hours. Determine the probability that a transistor will last for more than 400 hours.

25.

The grade point average of students at a university is normally distributed, with a mean of 2.6 and a standard deviation of 0.6. A recruiter for a company is interviewing students for summer employment. What percentage of the students will have a grade point average of 3.5 or greater?

26.

The weight of bags of fertilizer is normally distributed, with a mean of 50 pounds and a standard deviation of 6 pounds . What is the probability that a bag of fertilizer will weigh between 45 and 55 pounds ?

27.

The monthly demand for a product is normally distributed, with a mean of 700 units and a standard deviation of 200 units. What is the probability that demand will be greater than 900 units in a given month?

28.

The Polo Development Firm is building a shopping center. It has informed renters that their rental spaces will be ready for occupancy in 19 months. If the expected time until the shopping center is completed is estimated to be 14 months, with a standard deviation of 4 months, what is the probability that the renters will not be able to occupy in 19 months?

29.

A warehouse distributor of carpet keeps 6,000 yards of deluxe shag carpet in stock during a month. The average demand for carpet from the stores that purchase from the distributor is 4,500 yards per month, with a standard deviation of 900 yards. What is the probability that a customer's order will not be met during a month? (This situation is referred to as a stockout .)

30.

The manager of the local National Video Store sells videocassette recorders at discount prices. If the store does not have a video recorder in stock when a customer wants to buy one, it will lose the sale because the customer will purchase a recorder from one of the many local competitors . The problem is that the cost of renting warehouse space to keep enough recorders in inventory to meet all demand is excessively high. The manager has determined that if 90% of customer demand for recorders can be met, then the combined cost of lost sales and inventory will be minimized. The manager has estimated that monthly demand for recorders is normally distributed, with a mean of 180 recorders and a standard deviation of 60. Determine the number of recorders the manager should order each month to meet 90% of customer demand.

31.

The owner of Western Clothing Company has determined that the company must sell 670 pairs of denim jeans each month to break even (i.e., to reach the point where total revenue equals total cost). The company's marketing department has estimated that monthly demand is normally distributed, with a mean of 805 pairs of jeans and a standard deviation of 207 pairs. What is the probability that the company will make a profit each month?

32.

Lauren Moore, a professor in management science, is computing her final grades for her introductory management science class. The average final grade is a 63, with a standard deviation of 10. Professor Moore wants to curve the final grades according to a normal distribution so that 10% of the grades are Fs, 20% are Ds, 40% are Cs, 20% are Bs, and 10% are As. Determine the numeric grades that conform to the curve Professor Moore wants to establish.

33.

The SAT scores of all freshmen accepted at State University are normally distributed, with a mean of 1,050 and a standard deviation of 120. The College of Business at State University has accepted 620 of these freshmen into the college. All students in the college who score over 1,200 are eligible for merit scholarships. How many students can the college administration expect to be eligible for merit scholarships?

34.

Erin Richards is a junior at Central High School, and she has talked to her guidance counselor about her chances of being admitted to Tech after her graduation. The guidance counselor has told her that Tech generally accepts only those applicants who graduate in the top 10% of their high school class. The average grade point average of the last four senior classes has been 2.67, with a standard deviation of 0.58. What GPA will Erin have to achieve to be in the top 10% of her class?

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35.

The associate dean in the college of business at Tech is going to purchase a new copying machine for the college. One model he is considering is the Zerox X10. The sales representative has told him that this model will make an average of 125,000 copies, with a standard deviation of 40,000 copies, before breaking down. What is the probability that the copier will make 200,000 copies before breaking down?

36.

The Palace Hotel believes its customers may be waiting too long for room service. The hotel operations manager knows that the time for room service orders is normally distributed, and he sampled 10 room service orders during a 3-day period and timed each (in minutes), as follows:

 23 23 15 12 26 16 19 18 30 25

The operations manager believes that only 10% of the room service orders should take longer than 25 minutes if the hotel has good customer service. Does the hotel room service meet this goal?

37.

Agnes Hammer is a senior majoring in management science. She has been interviewing with several companies for a job when she graduates, and she is curious about what starting salary offers she might receive. There are 140 seniors in the graduating class for her major, and more than half have received job offers. She asked 12 of her classmates at random what their annual starting salary offers were, and she received the following responses:

 \$28,500 \$35,500 32,600 36,000 34,000 25,700 27,500 29,000 24,600 31,500 34,500 26,800

Assume that starting salaries are normally distributed. Compute the mean and standard deviation for these data and determine the probability that Agnes will receive a salary offer of less than \$27,000.

38.

The owner of Gilley's Ice Cream Parlor has noticed that she sells more ice cream on hotter days during the summer, especially on days when the temperature is 85 or higher. To plan how much ice cream to stock, she would like to know the average daily high temperature for the summer months of July and August. Assuming that daily temperatures are normally distributed, she has gathered the following data for the high temperature for 20 days from a local almanac:

 86 92 85 94 78 83 91 81 90 84 92 76 83 78 80 78 69 85 74 90

[Page 512]

Compute the mean and standard deviation for these data and determine the expected number of days in July and August that the high temperature will be 85 or greater.

39.

The state of Virginia has implemented a Standard of Learning (SOL) test that all public school students must pass before they can graduate from high school. A passing grade is 75. Montgomery County High School administrators want to gauge how well their students might do on the SOL test, but they don't want to take the time to test the whole student population. Instead, they selected 20 students at random and gave them the test. The results are as follows:

 83 79 56 93 48 92 37 45 72 71 92 71 66 83 81 80 58 95 67 78

Assume that SOL test scores are normally distributed. Compute the mean and standard deviation for these data and determine the probability that a student at the high school will pass the test.

40.

The department of management science at Tech has sampled 250 of its majors and compiled the following frequency distribution of grade point averages (on a 4.0 scale) for the previous semester:

GPA

Frequency

0 < 0.5

1

0.5 < 1.0

4

1.0 < 1.5

20

1.5 < 2.0

35

2.0 < 2.5

67

2.5 < 3.0

58

3.0 < 3.5

47

3.5 < 4.0

18

250

The sample mean ( ) for this distribution is 2.5, and the sample standard deviation ( s ) is 0.72. Determine whether the student GPAs are normally distributed, using a .05 level of significance (i.e., a = .05).

41.

Geo-net, a cellular phone company, has collected the following frequency distribution for the length of calls outside its normal customer roaming area:

Length (min.)

Frequency

0 < 5

26

5 < 10

75

10 < 15

139

15 < 20

105

20 < 25

37

25+

18

400

The sample mean ( ) for this distribution is 14.3 minutes, and the sample standard deviation is 3.7 minutes. Determine whether these data are normally distributed (

Introduction to Management Science (10th Edition)
ISBN: 0136064361
EAN: 2147483647
Year: 2006
Pages: 358

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