Practice problems from other books / resources Ex1. Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice. a. Construct the sample space and count the sample points. b. What is the probability of obtaining a value of sum of 7? c. What is the probability of obtaining a value of sum of 9 or greater? [Ans: a: 36, b: 6/36 c:10/36] Ex2: A small assembly plant has 50 employees. On occasion, some of the workers fail to meet the performance standards by completing work late or assembling a defective product. At the end of a performance evaluation period, the production manager found that 5 of the 50 workers completed work late, 6 of the 50 workers assembled a defective product, and 2 of the 50 workers both completed work late and assembled a defective product. The production manager decided to assign a poor performance rating to any employee whose work was either late or defective. What is the probability that the production manager assigned an employee a poor performance rating? [Ans: 9/50, addition law] Ex3: Clarkson University surveyed alumni to learn more about what they think of Clarkson.One part of the survey asked respondents to indicate whether their overall experience at Clarkson fell short of expectations, met expectations, or surpassed expectations. The results showed that 4% of the respondents did not provide a response, 26% said that their experience fell short of expectations, and 65% of the respondents said that their experience met expectations. a. If we chose an alumnus at random, what is the probability that the alumnus would say their experience surpassed expectations? b. If we chose an alumnus at random, what is the probability that the alumnus would say their experience met or surpassed expectations? [Ans: a. 0.05, b: 0.65+0.05 = 0.70, addition for ME events] Ex4: A survey of magazine subscribers showed that 45.8% rented a car during the past 12 months for business reasons, 54% rented a car during the past 12 months for personal reasons, and 30% rented a car during the past 12 months for both business and personal reasons. a. What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons? b. What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons ̅̅̅̅̅̅̅̅ [Ans: a: 0.698 (addition law), b:𝑃(𝐵 ∪ 𝑃) = 1 − 𝑃(𝐵 ∪ 𝑃) = 0.302] Ex5: Consider a service station manager who knows from past experience that 80% of the customers use a credit card when they purchase gasoline. What is the probability that the next two customers purchasing gasoline will each use a credit card? (what is your assumption in calculating this probability?). [Ans: 0.64,multiplication rule for independent events.] Ex6: The automobile industry sold 657,000 vehicles in the United States during January 2009. The Big Three U.S. automakers— General Motors, Ford, and Chrysler. A summary of sales by automobile manufacturer and type of vehicle sold is shown in the following table. Data are in thousands of vehicles. The non-U.S. manufacturers are led by Toyota, Honda, and Nissan. The category Light Truck includes pickup, minivan, SUV, and crossover models. a. If a vehicle was manufactured by one of the U.S. automakers, what is the probability that the vehicle was a car? What is the probability it was a light truck? b. If a vehicle was not manufactured by one of the U.S. automakers, what is the probability that the vehicle was a car? What is the probability it was a light truck? c. If the vehicle was a light truck, what is the probability that it was manufactured by one of the U.S. automakers? [Ans: a:0.3115, b: 0.6069, c:0.566, all conditional probabilities using reduced sample space] Ex7: In a survey of MBA students, the following data were obtained on “students’ first reason for application to the school in which they matriculated. a. If a student goes full time, what is the probability that school quality is the first reason for choosing a school? b. If a student goes part time, what is the probability that school quality is the first reason for choosing a school? c. Let A denote the event that a student is full time and let B denote the event that the student lists school quality as the first reason for applying. Are events A and B independent? Justify your answer. [Ans a: 0.473, b: 0.385, c: not independent(check if joint prob = product of marginal prob)] Ex8: Visa Card USA studied how frequently young consumers, ages 18 to 24, use plastic (debit and credit) cards in making purchases. The results of the study provided the following probabilities. The probability that a consumer uses a plastic card when making a purchase is 0.37. Given that the consumer uses a plastic card, there is a 0.19 probability that the consumer is 18 to 24 years old. Given that the consumer uses a plastic card, there is a 0.81 probability that the consumer is more than 24 years old. The U.S. Census Bureau data show that 14% of the consumer population is 18 to 24 years old. a. Given the consumer is 18 to 24 years old, what is the probability that the consumer use a plastic card? b. Given the consumer is over 24 years old, what is the probability that the consumer uses a plastic card? c. What is the interpretation of the probabilities shown in parts (a) and (b)? [Ans: a: 0.5021 (cond. prob rule), b: 0.3484(cond. prob rule), c: 50.21% of the young customers use plastic card for purchases, 34.84% of the more than 24 age customers use plastic cards for purchases.] Ex9:The Wall Street Journal/Harris Personal Finance poll asked 2082 adults if they owned a home (All Business website, January 23, 2008). A total of 1249 survey respondents answered Yes. Of the 450 respondents in the 18–34 age group, 117 responded Yes. a. What is the probability that a respondent to the poll owned a home? b. What is the probability that a respondent in the 18–34 age group owned a home? c. What is the probability that a respondent to the poll did not own a home? d. What is the probability that a respondent in the 18–34 age group did not own a home? [Ans: a: 0.5999, b: 0.26, c: 0.4001 d: 0.74] Ex10: A study of 31,000 hospital admissions in New York State found that 4% of the admissions led to treatment-caused injuries. One-seventh of these treatment-caused injuries resulted in death, and one-fourth of the treatment-caused injuries were caused by negligence. a. What is the probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence? b. What is the probability a person admitted to the hospital will die from a treatment caused injury? [Ans: a: 0.01, Multiplication rule, b: 1/175 = 0.00571(multiplication rule)] Ex11: A large consumer goods company ran a television advertisement for one of its soap products. On the basis of a survey that was conducted, probabilities were assigned to the following events. B: individual purchased the product S: individual recalls seeing the advertisement 𝐵 ∩ 𝑆: individual purchases the product and recalls the seeing the advertisement The probabilities assigned were P(B) = 0.20, P(S) = 0.40, and P(𝐵 ∩ 𝑆) =0 .12. a. What is the probability of an individual’s purchasing the product given that the individual recalls seeing the advertisement? Does seeing the advertisement increase the probability that the individual will purchase the product? As a decision maker, would you recommend continuing the advertisement (assuming that the cost is reasonable)? b. Assume that individuals who do not purchase the company’s soap product buy from its competitors. What would be your estimate of the company’s market share? Would you expect that continuing the advertisement will increase the company’s market share? Why or why not? c. The company also tested another advertisement and assigned it values of P(S) =0.30 and P(𝐵 ∩ 𝑆) = 0.10. What is P(B | S) for this other advertisement? Which advertisement seems to have had the bigger effect on customer purchases? [Ans a: 0.3 (cond. probrule) yes, b: 20% yes since P(B | S)> P(B) c: 0.333 other advertisement] Ex12: Cooper Realty is a small real estate company located in Albany, New York, specializing primarily in residential listings. They recently became interested in determining the likelihood of one of their listings being sold within a certain number of days. An analysis of company sales of 800 homes in previous years produced the following data. a.If A is defined as the event that a home is listed for more than 90 days before being sold, estimate the probability of A. b. If B is defined as the event that the initial asking price is under $150,000, estimate the probability of B. c. What is the probability of A ∩ B? d. Assuming that a contract was just signed to list a home with an initial asking price of less than $150,000, what is the probability that the home will take Cooper Realty more than 90 days to sell? e. Are events A and B independent? [Ans: a: 0.25 b: 0.125 c: 0.0125 (cond. prob rule) d: 0.1 e: no (check joint prob≠product of marginal prob] Ex13: A company knows that a rival is about to bring out a competing product. It believes that this rival has three possible packaging plans (superior, normal, and cheap) in mind and that all are equally likely. Also, there are three equally likely possible marketing strategies (intense media advertising, price discounts, and the use of a coupon to reduce the price of future purchases). What is the probability that the rival will employ superior packaging in conjunction with an intense media advertising campaign? Assume that packaging plans and marketing strategies are determined independently. [Ans: 1/9 (multiplication rule for independent events)] Ex14: A conference began at noon with two parallel sessions. The session on portfolio management was attended by 40% of the delegates, while the session on chartism was attended by 50%. The evening session consisted of a talk titled “Is the Random Walk Dead?” This was attended by 80% of all delegates. a. If attendance at the portfolio management session and attendance at the chartism session are mutually exclusive, what is the probability that a randomly chosen delegate attended at least one of these sessions? b. If attendance at the portfolio management session and attendance at the evening session are statistically independent, what is the probability that a randomly chosen delegate attended at least one of these sessions? c. Of those attending the chartism session, 75% also attended the evening session. What is the probability that a randomly chosen delegate attended at least one of these two sessions? [Ans: a: 0.9 (add. law for ME events), b: 0.88(add law with assumption for independence) c: 0.925(add law with intersection prob computed from cond. prob rule)] Ex15: A corporation was concerned with the basic educational skills of its workers and decided to offer a selected group of them separate classes in reading and practical mathematics. Of these workers, 40% signed up for the reading classes and 50% for the practical mathematics classes. Of those signing up for the reading classes 30% signed up for the mathematics classes. a. What is the probability that a randomly selected worker signed up for both classes? b. What is the probability that a randomly selected worker who signed up for the mathematics classes also signed up for the reading classes? c. What is the probability that a randomly chosen worker signed up for at least one of these two classes? d. Are the events “signs up for the reading classes” and “signs up for the mathematics classes” statistically independent? [Ans a: 0.12 (multiplication rule), b: 0.24 (cond. prob rule), c: 0.78 (add. law), d: no] Ex16: A consulting organization predicts whether corporations’ earnings for the coming year will be unusually low, unusually high, or normal. Before deciding whether to continue purchasing these forecasts, a stockbroker compares past predictions with actual outcomes. The accompanying table shows proportions in the nine joint classifications. a.What proportion of predictions have been for unusually high earnings? b. What proportion of outcomes has been for unusually high earnings? c. If a firm were to have unusually high earnings, what is the probability that the consulting organization would correctly predict this event? d. If the organization predicted unusually high earnings for a corporation, what is the probability that these would materialize? e. What is the probability that a corporation for which unusually high earnings had been predicted will have unusually low earnings? [Ans: a: 0.3, b: 0.38c: 0.605(cond. prob),d: 0.7666(cond. prob), e: 1/30 = 0.0333(cond. prob)] Ex17: Before books aimed at preschool children are marketed, reactions are obtained from a panel of preschool children. These reactions are categorized as favorable, neutral, or unfavorable. Subsequently, book sales are categorized as high, moderate, or low, according to the norms of this market. Similar panels have evaluated1,000 books in the past. The accompanying table shows their reactions and the resulting market performance of the books. a.If the panel reaction is favorable, what is the probability that sales will be high? b. If the panel reaction is unfavorable, what is the probability that sales will be low? c. If the panel reaction is neutral or better, what is the probability that sales will be low? d. If sales are low, what is the probability that the panel reaction was neutral or better? [Ans a: 0.5709(cond. prob), b: 0.5183(cond. prob), c: 0.2191(cond. prob), d:0.5236(cond. prob)] Ex18: An insurance company estimated that 30% of all automobile accidents were partly caused by weather conditions and that 20% of all automobile accidents involved bodily injury. Further, of those accidents that involved bodily injury, 40% were partly caused by weather conditions. a. What is the probability that a randomly chosen accident both was partly caused by weather conditions and involved bodily injury? b. Are the events “partly caused by weather conditions” and “involved bodily injury” independent? c. If a randomly chosen accident was partly caused by weather conditions, what is the probability that it involved bodily injury? d. What is the probability that a randomly chosen accident both was not partly caused by weather conditions and did not involve bodily injury? ̅̅̅̅̅̅̅̅̅̅ ̅ ) = 𝑃(𝐵 [Ans a: 0.08 b: no c:0.2666 d: Use De Morgan’s law: 𝑃(𝐵̅ ∩ 𝑊 ∪ 𝑊) = 1 − 𝑃(𝐵 ∪ 𝑊) ̅) = 1− 𝑃(𝐵 ∪ 𝑊) = 𝑃(𝐵) + 𝑃(𝑊) − 𝑃(𝐵(∩ 𝑊) = 0.2 + 0.3 − 0.08 = 0.42, 𝑃(𝐵̅ ∩ 𝑊 0.42 = 0.58] Ex19: In a campus restaurant it was found that 35% of all customers order vegetarian meals and that 50% of all customers are students. Further, 25% of all customers who are students order vegetarian meals. a. What is the probability that a randomly chosen customer both is a student and orders a vegetarian meal? b. If a randomly chosen customer orders a vegetarian meal, what is the probability that the customer is a student? c. What is the probability that a randomly chosen customer both does not order a vegetarian meal and is not a student? d. Are the events “customer orders a vegetarian meal” and “customer is a student” independent? e. Are the events “customer orders a vegetarian meal” and “customer is a student” mutually exclusive? [Ans a: 0.125 (multiplication rule), b: 0.3571(cond. prob) c: 0.275(Use DeMorgan’s law 𝑃(𝐴̅ ∩ 𝐵̅ ) = 1 − 𝑃(𝐴 ∪ 𝐵))], d: No e: No] Ex20: From past experience, a stockbroker believes that under present economic conditions a customer will invest in tax-free bonds with a probability of 0.6, will invest in mutual funds with a probability of 0.3, and will invest in both tax-free bonds and mutual funds with a probability of 0.15. At this time, find the probability that a customer will invest (a) in either tax-free bonds or mutual funds; (b) in neither tax-free bonds nor mutual funds. [Ans a: 0.75 b: 0.25 (Use DeMorgan’s law 𝑃(𝐴̅ ∩ 𝐵̅ ) = 1 − 𝑃(𝐴 ∪ 𝐵))] Ex21: According to Nielsen Media Research, approximately 67% of all U.S. households with television have cable TV. Seventy-four percent of all U.S. households with television have two or more TV sets. Suppose 55% of all U.S. households with television have cable TV and two or more TV sets. A U.S. household with television is randomly selected. a. What is the probability that the household has cable TV or two or more TV sets? b. What is the probability that the household has cable TV or two or more TV sets but not both? c.What is the probability that the household has neither cable TV nor two or more TV sets? [Ans a: 0.86 b:0.31 (use Venn diagram) c: 0.14 (use DeMorgan’s law 𝑃(𝐴̅ ∩ 𝐵̅ ) = 1 − 𝑃(𝐴 ∪ 𝐵)] Ex22: The following probability matrix contains a breakdown on the age and gender of U.S. physicians in a recent year, as reported by the American Medical Association. a. What is the probability that one randomly selected physician is 35–44 years old? b. What is the probability that one randomly selected physician is both a woman and 45–54 years old? c. What is the probability that one randomly selected physician is a man or is 35–44 years old? d. What is the probability that one randomly selected physician is less than 35 years old or 55–64 years old? e. What is the probability that one randomly selected physician is a woman if she is 45–54 years old? f. What is the probability that a randomly selected physician is neither a woman nor 55–64 years old? [Ans a:0.28, b:0.04 c:0.86 d:0.32 e: 0.1739 f:0.66, use De-Morgan’s law 𝑃(𝐴̅ ∩ 𝐵̅ ) = 1 − 𝑃(𝐴 ∪ 𝐵)] Ex23: The “random walk” theory of securities prices holds that price movements in disjoint time periods are independentof each other. Suppose that we record only whether the price is up or down each year, and that the probability that our portfolio rises in price in any one year is 0.65. (This probability is approximately correct for a portfolio containing equal dollar amounts of all common stocks listed on the New York Stock Exchange.) (a) What is the probability that our portfolio goes up for 3 consecutive years? (b) If you know that the portfolio has risen in price 2 years in a row, what probability do you assign to the event that it will go down next year? (c) What is the probability that the portfolio’s value moves in the same direction in both of the next 2 years? [Ans: a : 0.2746(multiplication rule for independent events) b: 0.35 (cond. prob keeping in view independence) c:0.545 (P(UP1∩UP2∩ UP3)+P(D1∩D2∩ D3)] Ex24: The probability that a man will be alive in 25 years in 3/5 and the probability that his wife will be alive in 25 years is 2/3. Find the probability that (a) both will be alive in 25 years (b) only man will be alive (c) only wife will be alive (d) neither will be alive. [Ans:a: 2/5 (multiplication rule assuming independence), b:1/5(multiplication rule assuming independence). Also if events are independent, their complements are also independent), c:4/15 (multiplication rule assuming independence) d: 2/15 (multiplication rule assuming independence)].
