Bayes’ Theorem
4.2 EXERCISES
Skill Development
4-26. A study was conducted in a school on how students travel to school. Following are the data collected for three methods students use to travel to school:
Methods Number of People
Carpool 35
Drive 14
Public transport 47
a. Construct a relative frequency table for the provided data.
b. What is the probability that a student is not driving to school?
c. What is the probability that a student either carpools or drives to school?
4-27. A Jack in the Box franchise manager has determined that the chance a customer will order a soft drink is 0.90. The probability that a customer will order a hamburger is 0.60. The probability that a customer will order french fries is 0.50.
a. If a customer places an order, what is the probability that the order will include a soft drink and no fries if these two events are independent?
b. The restaurant has also determined that if a customer orders a hamburger, the probability the customer will also order fries is 0.80. Determine the probability that the order will include a hamburger and fries.
4-28. Ponderosa Paint and Glass carries three brands of paint. A customer wants to buy another gallon of paint to match paint she purchased at the store previously. She can’t recall the brand name and does not wish to return home to find the old can of paint.
So she selects two of the three brands of paint at random and buys them.
a. What is the probability that she matched the paint brand?
b. Her husband also goes to the paint store and fails to remember what brand to buy. So he also purchases two of the three brands of paint at random.
Determine the probability that both the woman and her husband fail to get the correct brand of paint.
(Hint: Are the husband’s selections independent of his wife’s selections?)
4-29. A college basketball team has 10 players; 5 are seniors, 2 are juniors, and 3 are sophomores. Two players are randomly selected to serve as captains for the next game. What is the probability that both players selected are seniors?
4-30. There are reasons for students to choose specific universities for higher studies. In a survey of
undergraduate students, the following data were obtained on students applying a degree program in a university based on their enrollment status:
Enrollment Status University
Quality Program
Fees Other
Full time 42 39 75
Part time 40 58 46
If a student is selected at random, find the probability that
a. a student enrolls as full-tim
e
student.
b. the student applies as a full-time student because of the quality of the university
.
c. the student applies as a part-time student because of the program’s fees.
4-31. Three events occur with probabilities
P1E12 = 0.35, P1E22 = 0.15, P1E32 = 0.40. If the event B occurs, the probability becomes
P1E1B2 = 0.25, P1B2 = 0.30.
a. Calculate P1E1 and B2. b. Compute P1E1 or B2.
c. Assume that E1, E2, and E3 are independent events.
Calculate P1E1 and E2 and E32.
4-32. The URS Corporation construction company has submitted two bids, one to build a large hotel in London and the other to build a commercial office building in New York City. The company believes it has a 40% chance of winning the hotel bid and a 25% chance of winning the office building bid.
The company also believes that winning the hotel bid is independent of winning the office building bid.
a. What is the probability the company will win both contracts?
b. What is the probability the company will win at least one contract?
c. What is the probability the company will lose both contracts?
4-33. Suppose a quality manager for Dell Computers has collected the following data on the quality status of disk drives by supplier. She inspected a total of 700 disk drives.
Drive Status Supplier Working Defective
Company A 120 10
Company B 180 15
Company C 50 5
Company D 300 20
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Chapter 4 1854-37. Suppose the managers at FedEx Office wish to meet the increasing demand for color photocopies and to have more reliable service. (Refer to Exercise 4-36.) As a goal, they would like to have at least a 99.9% chance of being able to furnish a black-and-white copy or a color copy on demand. They also wish to purchase only four copiers. They have asked for your advice regarding the mix of black-and-white and color copiers. Supply them with your advice. Provide calculations and reasons to support your advice.
4-38. Cindy is planning for a day trip and is deciding whether to wear closed-toed shoes or sandals during her trip. She is also checking the weather reports. If the report indicates it will rain then she will wear closed- toed shoes; otherwise, she will wear a pair of sandals.
The weather conditions state that 36 of 60 days will have similar conditions. The last five days’ weather report showed that three days would be sunny and two days would have rain. Use probability to determine the weather for Cindy’s day trip and which shoes she will need to wear.
4-39. RAC is an automotive dealer selling imported new cars, and a number of them are Japanese cars. The manager of RAC found that 60% of the customers visited them but didn’t purchase anything. About 25%
bought an imported new car and 45% bought a
Japanese car. What is the probability a customer visited them and bought an imported new Japanese car?
4-40. A government-sponsored subsidized housing
construction contract is to be awarded on the basis of a blind drawing from those who have bid. In addition to the Hubble Company, five other companies have submitted bids.
a. What is the probability of the Hubble Construction Company winning the bid?
b. Suppose two contracts are to be awarded by a blind draw. What is the probability of Hubble winning both contracts? Assume sampling with replacement.
c. Referring to part b, what is the probability of Hubble not winning either contract?
d. Referring to part b, what is the probability of Hubble winning exactly one contract?
4-41. Dreamland has its own factory in manufacturing pillows. It manufactures two types of pillows—
polyester pillows and memory foam pillows.
Dreamland distributes its pillows to hospitals, hotels, and department stores. The following are the number of distributions made in a day:
Pillow Type Hospitals Hotels Department Stores
Polyester pillows 42 74 154
Memory foam pillows 34 21 131
a. Based on these inspection data, what is the probability of randomly selecting a disk drive from Company B?
b. What is the probability of a defective disk drive being received by the computer company?
c. What is the probability of a defect given that Company B supplied the disk drive?
4-34. Three events occur with probabilities P1E12 = 0.35, P1E22 = 0.25, P1E32 = 0.40. Other probabilities are P1BE12 = 0.25, P1BE22 = 0.15, P1BE32 = 0.60.
a. Compute P1E1B2. b. Compute P1E2B2. c. Compute P1E3B2.
4-35. A hair salon caters to both men and women. In addition to hairstyles, the salon provides back and shoulder massages. In the past, 42% of men have requested a massage and 61% of women have
requested a massage. The salon’s customers are 48.2%
men and 51.8% women.
a. If a customer arrives for a hair appointment, what is the probability that the customer is a woman who will also request a massage?
b. Calculate the probability that a customer will ask for a massage.
c. Given that the customer does ask for a massage, what is the probability that the customer is a man?
Business Applications
4-36. A local FedEx Office has three black-and-white copy machines and two color copiers. Based on historical data, the chance that each black-and-white copier will be down for repairs is 0.10. The color copiers are more of a problem and are down 20% of the time each.
a. Based on this information, what is the probability that if a customer needs a color copy, both color machines will be down for repairs?
b. If a customer wants both a color copy and a black- and-white copy, what is the probability that the necessary machines will be available? (Assume that the color copier can also be used to make a black- and-white copy if needed.)
c. If the manager wants to have at least a 99% chance of being able to furnish a black-and-white copy on demand, is the present configuration sufficient?
(Assume that the color copier can also be used to make a black-and-white copy if needed.) Back up your answer with appropriate probability
computations.
d. What is the probability that all five copiers will be up and running at the same time? Suppose the manager added a fourth black-and-white copier.
How would the probability of all copiers being ready at any one time be affected?
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Introduction to Probabilityprobability 0.34. The chance of receiving a signal given using the new model transmitter is 80%; there is 77%
chance of receiving a signal given using the old transmitter. What is the probability that a signal is received? (Hint: Use Bayes’ rule)
4-46. A major electronics manufacturer has determined that when one of its televisions is sold, there is 0.08 chance that it will need service before the warranty period expires. It has also assessed a 0.05 chance that a digital video recorder (DVR) will need service prior to the expiration of the warranty.
a. Suppose a customer purchases one of the company’s televisions and one of the DVRs.
What is the probability that at least one of the products will require service prior to the warranty expiring?
b. Suppose a retailer sells four televisions on a particular Saturday. What is the probability that none of the four will need service prior to the warranty expiring?
c. Suppose a retailer sells four televisions on a particular Saturday. What is the probability that at least one will need repair?
4-47. Uncle John owns a roadside magazine stall. This morning it starts to rain. Given a rainy morning, the chance uncle John sold all his newspapers is 25%.
When it is not a rainy morning, uncle John would have sold 85% of all his newspapers. The probability of a rainy morning is 0.125.
a. What is the probability uncle John sold all his newspapers and it is a rainy morning?
b. What is the probability it is not a rainy morning and uncle John sold all his newspapers?
4-48. A distributor of outdoor yard lights has four suppliers.
This past season, she purchased 40% of the lights from Franklin Lighting, 30% from Wilson & Sons, 20%
from Evergreen Supply, and the rest from A. L. Scott.
In prior years, 3% of Franklin’s lights were defective, 6% of the Wilson lights were defective, 2% of Evergreen’s were defective, and 8% of the Scott lights were defective. When the lights arrive at the distributor, she puts them in inventory without identifying the supplier. Suppose that a defective light string has been pulled from inventory; what is the probability that it was supplied by Franklin Lighting?
4-49. Your company assigned two accident insurance plans (Plan 1 and Plan 2) among the full-time and part-time staffs. There are 70% of full-time staffs in the company and 60% of staffs had been assigned in Plan 2. Given Plan 1, the probability for full-time staff is 0.75 and the probability for part-time staff is 0.25. The company assigned Plan 2 to part-time staffs with the probability of 0.2.
a. What is the probability of a full-time staff with Plan 1 accident insurance?
a. What is the probability that pillows were sent to hotels?
b. What is the probability that memory foam pillows were sent to department stores?
c. A polyester pillow was sent out; what is the probability it was sent to a hospital?
d. Determine whether the pillow types are independent from the places they are sending to.
4-42. Online marketplaces such as Craigslist have become a popular way for individuals to buy and sell
miscellaneous items. The table below shows the numbers of days products stayed active (not sold) on one of these sites and also the price range of the items.
Days on the Website Price Range 1-7 8-30 Over 30
Under $200 125 15 30
$200–$500 200 150 100
$501–$1,000 400 525 175
Over $1,000 125 140 35
a. Using the relative frequency approach to probability assessment, what is the probability that a product will be on the website more than 7 days?
b. Is the event 1-7 days on the website independent of the price $200–$500?
c. Suppose an item has just sold and was on the website less than 8 days, what is the most likely price range for that item?
4-43. Vegetables from the summer harvest are processed at a food-processing plant. The manager has found a case of cans that have not been properly sealed. There are three lines that processed cans of this type, and the manager wants to know which line is most likely to be responsible for this mistake. Provide the manager this information.
Line Contribution
to Total Proportion Defective
1 0.40 0.05
2 0.35 0.10
3 0.25 0.07
4-44. A corporation has 11 manufacturing plants. Of these, seven are domestic and four are outside the United States. Each year a performance evaluation is conducted for four randomly selected plants. What is the probability that a performance evaluation will include at least one plant outside the United States?
(Hint: Begin by finding the probability that only domestic plants are selected.)
4-45. Nikon is launching their new wireless transmitter, which implemented better send and receive technology.
The signal is transmitted using the new model with probability 0.76 and using the old model with
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4.2 The Rules of Probability
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Chapter 4 187b. What is the probability of a part-time staff with Plan 1 or Plan 2?
c. If two staffs were selected, what is the probability of both of them assigned with Plan 1?
4-50. Suppose an auditor has 18 tax returns, 12 of which are for physicians. If three of the 18 tax returns are randomly selected, then what is the probability that at least one of the three selected will be a physician’s tax return?
4-51. A box of 50 remote control devices contains three that have a defective power button. If devices are randomly sampled from the box and inspected one at a time, determine
a. The probability that the first control device is defective.
b. The probability that the first control device is good and the second control device is defective.
c. The probability that the first three sampled devices are all good.
Computer Software Exercises
4-52. ECCO is a company that makes backup alarms for commercial vehicles and heavy-duty equipment like forklifts. The data file ECCO contains warranty claim data as well as the associated manufacturing plant, complaint code, and shift for each alarm that is manufactured.
a. Based on the data provided, what is the probability that a warranty claim comes from a product made in the Salt Lake City plant?
b. What is the probability that a warranty claim was made on the swing shift at the Toronto plant?
c. What is the probability that a warranty claim is for a cracked lens or for a wiring complaint?
d. If two warranty claims are examined, what is the probability that both are from a product made in Boise?
4-53. A hotel chain conducted a survey of its guests. Sixty- two surveys were completed. Based on the data from the survey, found in the file HotelSurvey, answer the following questions using the relative frequency assessment method.
a. Of two customers selected, what is the probability that both will be on a business trip?
b. What is the probability that a customer will be on a business trip or will experience a hotel problem during a stay at the hotel?
c. What is the probability that a customer on business has an in-state area code phone number?
d. Based on the data in the survey, can the hotel manager conclude that a customer’s rating regarding staff attentiveness is independent of whether he or she is traveling on business, pleasure, or both? Use the rules of probability to make this determination.
4-54. Continuing with the survey done by the managers of the hotel chain, based on the data from the survey, found in the file HotelSurvey, determine the following probabilities using the relative frequency assessment method.
a. Of two customers selected, what is the probability that neither will be on a business trip?
b. What is the probability that a customer will be on a business trip or will not experience a hotel problem during a stay at the hotel?
c. What is the probability that a customer on a pleasure trip has an in-state area code phone number?
4-55. A survey asked what the most important reason was that people give for not using a wireless phone exclusively. The responses were: (1) Like the safety of traditional phone, (2) Like having a land line for a backup phone, (3) Pricing not attractive enough, (4) Weak or unreliable cell signal at home, (5) Need phone line for DSL Internet access, and (6) Other. The file titled Wireless contains the responses for 1,088 respondents.
a. Of those respondents 36 or older, determine the probability that an individual in this age group would not use a wireless phone exclusively because of concern about a weak or unreliable cell signal.
b. If three respondents were selected at random from those respondents younger than 36, calculate the probability that at least one of the respondents stated the most important reason for not using a wireless exclusively was that he or she liked having a land line for a backup phone.
4-56. The loan manager for High Desert Bank selected a sample of the loans her bank has made in the past year.
The file High Desert Banking contains information on the loan amount, the type of loan, and the region of the state where the loan was made.
a. What is the probability of randomly selecting a loan from this file that has a loan amount of at least
$73,500 and was made in Region 2?
b. Find the probability that a commercial loan is less than $90,000.
c. If two loans are selected, what is the probability that both are commercial loans?
4-57. A report indicated the average number of days it takes for an American company to fill a job vacancy is 48 days. Sample data similar to those used in the study are in a file titled Hired. Categories for the days and hire cost are provided under the headings “Time” and
“Cost,” respectively.
a. Calculate the probability that a company vacancy took at most 100 days or cost at most $4,000 to fill.
b. Of the vacancies that took at most 100 days to fill, calculate the probability that the cost was at most
$4,000.
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Introduction to Probabilitya. If a scooter was assembled in the Tyler plant, what is the probability its breakdown was due to an electrical problem?
b. Is the probability of a scooter having a mechanical problem independent of the scooter being
assembled at the Lincoln plant?
c. If mechanical problems are assigned a cost of $75 and electrical problems are assigned a cost of $100, how much cost would be budgeted for the Lincoln and Tyler plants next year if a total of 500 scooters were expected to be returned for repair?
c. If three of the vacancies were chosen at random, calculate the probability that two of the vacancies cost at most $4,000 to fill.
4-58. A scooter manufacturing company is notified whenever a scooter breaks down, and the problem is classified as being either mechanical or electrical. The company then matches the scooter to the plant where it was assembled. The file Scooters contains a random sample of 200 breakdowns. Use the data in the file to answer these questions.
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4 Overview
• Probability is the way we measure uncertainty about events. To properly use probability, you need to know the probability rules and the terms associated with probability.
• Important probability concepts and terminology include sample space, dependent and independent events, and mutually exclusive events.
• Probabilities are assessed in three main ways: classical assessment, relative frequency assessment, and subjective assessment.
o u t c o m e 1 Identify situations for which each of the three approaches to assessing probabilities applies.
The Basics of Probability (pg. 153–165)
4.1
Summary
• Section 4.2 introduces nine probability rules, including three addition rules and two multiplication rules.
• Rules for conditional probability and the complement rule are also very useful.
• Bayes’ Theorem is used to calculate conditional probabilities in situations where the probability of the given event is not provided and must be calculated. Bayes’
Theorem is also used to revise prior probabilities after new information affecting the prior becomes available.