1. Use the values in the cross-tabulation table to solve the equations given. Variable 1 D E A 10 20 Variable 2 B 15 5 C 30 15 a. P(E) b. P(B U D) c. P(A E) d. P(B|E) e. P(A U B) f. P(B C) g. P(D|C) h. P(A|B) i. Are variables 1 and 2 independent? Why or why not? 2. Use the values in the cross-tabulation table to solve the equations given. D E F G A 3 9 7 12 B 8 4 6 4 C 10 5 3 7 a. P(F A) b. P(A|B) c. P(B) d. P(E F) e. P(D|B) f. P(B|D) g. P(D U C) h. P(F) 3. Purchasing Survey asked purchasing professionals what sales traits impressed them most in a sales representative. Seventy-eight percent selected "thoroughness." Forty percent responded, "knowledge of your own product." The purchasing professionals were allowed to list more than one trait. Suppose 27% of the purchasing professionals listed both "thoroughness" and "knowledge of your own product" as sales traits that impressed them most. A purchasing professional is randomly sampled. a. What is the probability that the professional selected "thoroughness" or "knowledge of your own product"? b. What is the probability that the professional selected neither "thoroughness" nor "knowledge of your own product"? c. If it is known that the professional selected "thoroughness," what is the probability that the professional selected "knowledge of your own product"? d. What is the probability that the professional did not select "thoroughness" and did select "knowledge of your own product"? 4. The U.S. Bureau of Labor Statistics publishes data on the benefits offered by small companies to their employees. Only 42% offer retirement plans while 61% offer life insurance. Suppose 35% offer both retirement plans and life insurance as benefits. If a small company is randomly selected, determine the following probabilities: a. The company offers a retirement plan given that they offer life insurance. b. The company offers life insurance given that they offer a retirement plan. c. The company offers life insurance or a retirement plan. d. The company offers a retirement plan and does not offer life insurance. e. The company does not offer life insurance if it is known that they offer a retirement plan. 5. An Adweek Media/Harris Poll revealed that 44% of U.S. adults in the 18-34 years category think that "Made in America" ads boost sales. A different Harris Interactive poll showed that 78% of U.S. adults in the 18-34 years category use social media online. Suppose that 85% of U.S. adults in the 18-34 years category think that "Made in America" ads boost sales or use social media online. If a U.S. adult in the 18-34 years category is randomly selected, a. What is the probability that the person thinks that "Made in America" ads boost sales and uses social media online? b. What is the probability that the person neither thinks that "Made in America" ads boost sales nor uses social media online? c. What is the probability that the person thinks that "Made in America" ads boost sales and does not use social media online? d. What is the probability that the person thinks that "Made in America" ads boost sales given that the person does not use social media online? e. What is the probability that the person either does not think that "Made in America" ads boost sales or does use social media online? 6. In a certain city, 35% of the families have a MasterCard, 22% have an American Express card, and 28% have a Visa card. Eight percent of the families have both a MasterCard and an American Express card. Twelve percent have both a Visa card and a MasterCard. Six percent have both an American Express card and a Visa card. a. What is the probability of selecting a family that has either a Visa card or an American Express card? b. If a family has a MasterCard, what is the probability that it has a Visa card? c. If a family has a Visa card, what is the probability that it has a MasterCard? d. Is possession of a Visa card independent of possession of a MasterCard? Why or why not? e. Is possession of an American Express card mutually exclusive of possession of a Visa card? 7. A few years ago, a survey commissioned by The World Almanac and Maturity News Service reported that 51% of the respondents did not believe the Social Security System will be secure in 20 years. Of the respondents who were age 45 or older, 70% believed the system will be secure in 20 years. Of the people surveyed, 57% were under age 45. One respondent is selected randomly. a. What is the probability that the person is age 45 or older? b. What is the probability that the person is younger than age 45 and believes that the Social Security System will be secure in 20 years? c. If the person selected believes that Social Security System will be secure in 20 years, what is the probability that the person is 45 years old or older? d. What is the probability that the person is younger than age 45 or believes the Social Security System will not be secure in 20 years? 8. A telephone survey conducted by the Maritz Marketing Research company found that 43% of American expect to save money next year than they saved last year. Forty-five percent of those surveyed plan to reduce debt next year. Of those who expect to save more money next year, 85% plan to reduce debt next year. An American is selected randomly, a. What is the probability that this person expects to save more money next year and plans to reduce debt next year? b. What is the probability that this person expects to save more money next year or plans to reduce debt next year? c. What is the probability that this person neither expects to save more money next year nor plans to reduce debt next year? d. What is the probability that this person expects to save more money next year and does not plan to reduce debt next year? 9. The Steelcase Workplace Index studied the types of work-related activities that Americans did while on vacation in the summer. Among other things, 40% read work-related material. Thirty-four percent checked in with the boss. Respondents to the study were allowed to select more than one activity. Suppose that of those who read work-related material, 78% checked in with the boss. One of these survey respondents is selected randomly. a. What is the probability that while on vacation this respondent checked in with the boss and read work-related material? b. What is the probability that while on vacation this respondent neither checked in with the boss nor read work-related material? c. What is the probability that while on vacation this respondent read work-related material given that the respondent checked in with the boss? d. What is the probability that while on vacation this respondent did not check in with the boss given that the respondent read work-related material? e. What is the probability that while on vacation this respondent did not check in with the boss given that the respondent did not read work-related material? f. Construct a joint probability table for this problem. 10. A study on ethics in the workplace by the Ethics Resource Center and Kronos, Inc., revealed that 35% of employees admit to keeping quiet when they see coworker misconduct. Suppose that 80% of employees who admit to keeping quiet when they see coworker misconduct call in sick when they are well. In addition, suppose that 38% of employees who call in sick when they are well admitted to keeping quiet when they see coworker misconduct. If an employee is randomly selected, determine the following probabilities: a. The employee calls in sick when well and admits to keeping quiet when seeing coworker misconduct. b. The employee admits to keeping quiet when seeing coworker misconduct or calls in sick when well. c. Given that the employee calls in sick when well, he or she does not keep quiet when seeing coworker misconduct. d. The employee neither keeps quiet when seeing coworker misconduct nor calls in sick when well. e. The employee admits to keeping quiet when seeing coworker misconduct and does not call in sick when well. 11. Health Rights Hotline published the results of a survey of 2,400 people of Northern California in which consumers were asked to share their complaints about managed care. The number one complaint was denial of care, with 17% of the participating consumers selecting it. Several other complaints were noted, including inappropriate care (14%), customer service (14%), payment disputes (11%), specialty care (10%), delays in getting care (8%), and prescription drugs (7%). These complaint categories are mutually exclusive. Assume that the results of this survey can be inferred to all managed care consumers. If a managed care consumer is randomly selected, determine the following probabilities: a. The consumer complains about payment disputes or specialty care. b. The consumer complains about prescription drugs and customer service. c. The consumer complains about inappropriate care given that the consumer complains about specialty care. d. The consumer does not complain about delays in getting care nor does the consumer complain about payment disputes. 12. Companies use employee training for various reasons, including employee loyalty, certification, quality, and process improvement. In a national survey of companies, BI Learning Systems reported that 58% of the responding companies named employee retention as a top reason for training. Suppose 38% of the companies replied that they use training for process improvement and for employee retention. In addition, suppose that of the companies that use training for process improvement, 91% use training for employee retention. A company that used training is randomly selected. a. What is the probability that the company uses training for employee retention and not for process improvement? b. If it is known that the company uses training for employee retention, what is the probability that it uses training for process improvement? c. What is the probability that the company uses training for process improvement? d. What is the probability that the company uses training for employee retention or process improvement? e. What is the probability that the company neither uses training for employee retention nor uses training for process improvement? f. Suppose that it is known that the company does not use training for process improvement. What is the probability that the company does use training for employee retention? 13. Pitney Brown surveyed 302 directors and vice presidents of marketing at large and midsize U.S. companies to determine what they believe is the best vehicle for educating decision makers on complex issues in selling products and services. The highest percentage of companies chose direct mail/catalogs, followed by direct sales/sales rep. Direct mail/catalogs was selected by 38% of the companies. None of the companies selected both direct mail/catalogs and direct sales/sales rep. Suppose also that 41% selected neither direct mail/catalogs nor direct sales/sales rep. If one of these companies is selected randomly and their top marketing person interviewed about this matter, determine the following probabilities: a. The marketing person selected direct mail/catalogs and did not select direct sales/sales rep. b. The marketing person selected direct sales/sales rep. c. The marketing person selected direct sales/sales rep. given that the person selected direct mail/catalogs. d. The marketing person did not select direct sales/sales rep. given that the person does not select direct mail/catalogs. 14. In a study of incentives used by companies to retain mature workers by the The Conference Board, it was reported that 40% use flexible work arrangements. Suppose that of those companies that do not use flexible work arrangements, 11% give time off for volunteerism. In addition, suppose that of those companies that use flexible work arrangements, 62% give time off for volunteerism. If a company is randomly selected, determine the following probabilities: a. The company uses flexible work arrangements or gives time off for volunteerism. b. The company uses flexible work arrangements and does not give time off for volunteerism. c. Given that the company does not give time off for volunteerism, the company uses flexible work arrangements. d. The company does not use flexible work arrangements given that the company does give time off for volunteerism. e. The company does not use flexible work arrangements, or the company does not give time off for volunteerism. 15. A small independent physicians' practice has three doctors. Dr. Sarabia sees 41% of the patients, Dr. Tran sees 32%, and Dr. Jackson sees the rest. Dr. Sarabia requests blood tests on 5% of her patients, Dr. Tran requests blood tests on 8% of his patients, and Dr. Jackson requests blood tests on 6% of her patients. An auditor randomly selects a patient from the past week and discovers that the patient had a blood test as a result of the physician visit. Knowing this information, what is the probability that the patient saw Dr. Sarabia? For what percentage of all patients at this practice are blood tests requested? 16. A survey by the Arthur Andersen Enterprise Group/National Small Business United attempted to determine what the leading challenges are for the growth and survival of small businesses. Although the economy and finding qualified workers were the leading challenges, several others were listed in the results of the study, including regulations, listed by 30% of the companies, and the tax burden, listed by 35%. Suppose that 75% of the companies listing regulations as a challenge listed tax burden as a challenge. Assume these percentages hold for all small businesses. If a small business is randomly selected, determine the following probabilities: a. The small business lists both the tax burden and regulations as a challenge. b. The small business lists either the tax burden or regulations as a challenge. c. The small business lists either the tax burden or regulations but not both as a challenge. d. The small business lists regulations as a challenge given that it lists the tax burden as a challenge. e. The small business does not list regulations as a challenge given that it lists the tax burden as a challenge. f. The small business does not list regulations as a challenge given that it does not list the tax burden as a challenge. 17. According to U.S. Census Bureau figures, 35.3% of all Americans are in the 0-24 age bracket 14.2% are in the 25-34 age bracket, 16.0% are in the 35-44 age bracket, and 34.5% are in the 45 and older age bracket. A study by Jupiter Media Metrix determined that Americans use their leisure time in different ways according to age. For example, of those who are in the 45 and older age bracket, 39% read a book or a magazine more than 10 hours per week. Of those who are in the 0-24 age bracket, only 11% read a book or a magazine more than 10 hours per week. The percentage figures for reading a book or a magazine for more than 10 hours per week are 24% for the 25-34 age bracket and 27% for the 35-44 age bracket. Suppose an American is randomly selected and it is determined that he or she reads a book or a magazine more than 10 hours per week. Revise the probabilities that he or she is in any given age category. Using these figures, what is the overall percentage of the U.S. population that reads a book or a magazine more than 10 hours per week? 18. A retail study by Deloitte revealed that 55% of adults surveyed believed that plastic, non-compostable shopping bags should be banned. Suppose 40% of adults regularly recycle aluminum cans and believe that plastic, non-compostable shopping bags should be banned. In addition, suppose that 60% of adults who do not believe that plastic, non-compostable shopping bags should be banned do recycle. If an adult is randomly selected, a. What is the probability that the adult recycles and does not believe that plastic, non-compostable shopping bags should be banned? b. What is the probability that the adult does recycle? c. What is the probability that the adult does recycle or does believe that plastic, non-compostable shopping bags should be banned? d. What is the probability that the adult does not recycle or does not believe that plastic, non-compostable shopping bags should be banned? e. What is the probability that the adult does not believe that plastic, non-compostable shopping bags should be banned given that the adult does recycle? 19. A large international retailer has sourcing agents worldwide to search out suppliers of products according to the best quality/price ration for products that it sells in its stores in the United States. The retailer has a total of 131 sourcing agents internationally. Of these 51 specialize in textiles, 32 in footwear, and 17 in both textiles and footwear. The remainder are general sourcing agents with no specialization. All the sourcing agents are in a general database with a common E-mail address. When a purchasing manager from any of the retail stores needs information on its sourced products, they send an E-mail to the general database address. Anyone of the 131 sourcing agents can respond to the E-mail. a. Illustrate the category of the specialization of the sourcing agents on a Venn diagram. b. What is the probability that at any time an E-mail is sent it will be received by a sourcing agent specializing in textiles? c. What is the probability that at any time an E-mail is sent it will be received by a sourcing agent specializing in both textiles and footwear? d. What is the probability that at any time an E-mail is sent it will be received by a sourcing agent with no specialty? e. Given that the E-mail is received by a sourcing agent specializing in textiles what is the probability that the agent also has a specialty in footwear? f. Given that the E-mail is received by a sourcing agent specializing in footwear what is the probability that the agent also has a specialty in textiles? 20. George is an engineer in a design company. When the weather is nice, he walks to work and sometimes he cycles. In bad weather he takes the bus or he drives. Based on past habits there is a 10% probability that George walks, 30% he uses his bike, 20% he drives, and 40% of the time he takes the bus. If George walks, there is a 15% probability of being late to the office, if he cycles there is 10% chance of being late, a 55% chance of being late if he drives, and a 20% chance of being late if he takes the bus. a. On any given day, what is the probability of George being late to work? b. Given that George is late 1 day, what is the probability that he drove? c. Given that George is on time for work 1 day, what is the probability that he walked? d. Given that George takes the bus 1 day, what is the probability that he will arrive on time? e. Given that George walks to work 1 day, what is the probability that he will arrive on time?