Important Probability Terms
4.1 EXERCISES
4-4. Use a tree diagram to list the sample space for the number of movies rented by three customers at a Red Box video kiosk where customers are allowed to rent one, two, or three movies (assuming that each customer rents at least one movie).
4-5. In each of the following, indicate what method of probability assessment would most likely be used to assess the probability.
a. What is the probability that a major earthquake will occur in California in the next three years?
b. What is the probability that a customer will return a purchase for a refund?
c. An inventory of appliances contains four white washers and one black washer. If a customer selects one at random, what is the probability that the black washer will be selected?
Skill Development
4-1. You toss a fair coin in the air and then draw a card from a deck of 52 cards. Will the probability of drawing a king from the deck be affected by the output of the coin toss?
4-2. In a survey, respondents were asked to indicate their favorite brand of cereal (Post or Kellogg’s). They were allowed only one choice. What is the probability concept that implies it is not possible for a single respondent to state both Post and Kellogg’s are the favorite cereal?
4-3. If two customers are asked to list their choice of ice cream flavor from among vanilla, chocolate, and strawberry, list the sample space showing the possible outcomes.
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4.1 The Basics of Probability
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Chapter 4 1634-6. One of your classmates lives in the same locality where you stay. One of the classmate’s car broke down yesterday. You suspect there is a 90% of chance that she will call you for a ride to school today as you are the only one who resides near her.
a. Which probability assessment approach did you use for the given situation? Explain.
b. Is the relative frequency assessment approach applicable for this situation? State your reason.
4-7. Students who live on campus and purchase a meal plan are randomly assigned to one of three dining halls: the Commons, Northeast, and Frazier. What is the probability that the next student to purchase a meal plan will be assigned to the Commons?
4-8. The results of a census of 2,500 employees of a mid- sized company with 401(k) retirement accounts are as follows:
Account Balance
(to nearest $) Male Female 6+25,000 635 495 +25,000-+49,999 185 210 +50,000-+99,999 515 260
Ú+100,000 155 45
Suppose researchers are going to sample employees from the company for further study.
a. Based on the relative frequency assessment method, what is the probability that a randomly selected employee will be a female?
b. Based on the relative frequency assessment method, what is the probability that a randomly selected employee will have a 401(k) account balance between $25,000 and $49,999?
c. Compute the probability that a randomly selected employee will be a female with an account balance between $50,000 and $99,999.
4-9. Cross County Bicycles makes two mountain bike models, the XB-50 and the YZ-99, in three distinct colors. The following table shows the production volumes for last week:
Color
Model Blue Brown White
XB-50 302 105 200
YZ-99 40 205 130
a. Based on the relative frequency assessment method, what is the probability that a mountain bike is brown?
b. What is the probability that the mountain bike is a YZ-99?
c. What is the joint probability that a randomly selected mountain bike is a YZ-99 and brown?
d. Suppose a mountain bike is chosen at random.
Consider the following two events: the event that model YZ-99 is chosen and the event that a white product is chosen. Are these two events mutually exclusive? Explain.
4-10. Cyber-Plastics, Inc., is in search of a CEO and a CFO.
The company has a short list of candidates for each position. The four CEO candidates graduated from Chicago (C) and three Ivy League universities:
Harvard (H), Princeton (P), and Yale (Y). The four CFO candidates graduated from MIT (M),
Northwestern (N), and two Ivy League universities:
Dartmouth (D) and Brown (B). One candidate from each of the respective lists will be chosen randomly to fill the positions. The event of interest is that both positions are filled with candidates from the Ivy League.
a. Determine whether the outcomes are equally likely.
b. Determine the number of equally likely outcomes.
c. Define the event of interest.
d. Determine the number of outcomes associated with the event of interest.
e. Compute the classical probability of the event of interest using Equation 4.1.
4-11. Tomorrow, Sharon is going to sit for her mathematics test. The test paper consists 10 multiple choice questions with four possible answers A, B, C, and D for each question. Due to lack of practice, she is going to try her luck by guessing the correct answer for each question. Sharon needs to pass the test by getting at least five answers that are correct, but she is expecting correctly guessed five answers to pass the exam.
a. Determine whether the possible answers A, B, C, and D are equally likely.
b. Determine the total number of equally likely outcomes for all the questions.
c. Define the event of interest for Sharon.
d. Determine the number of outcomes with the event of interest for Sharon.
e. Compute the classical probability using Equation 1.
Business Applications
4-12. Shaun follows his friend to the horse betting field. This is his first time there and he has no experience on how to bet in horse races. He makes his choice based on his friend’s information and the data showed on the display board. What probability assessment approach did Shaun use? Explain.
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164 Chapter 4
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Introduction to Probability4-13. It is a sunny day and Cecelia goes to a cafe to purchase an ice-cream. In terms of size, she has three choices—small, medium, or large. For each size she chooses, she may include one topping of chocolate chips, wild strawberries, or almond flakes. By using a tree diagram, help Cecelia determine how many different choices of ice-cream she has.
4-14. China’s stock markets are tumbling. A three week’s plunge has knocked off about 5%, 6%, and 7% weekly Chinese shares. Investors are concerned and many of them have borrowed to be able to continue playing in the stock market. Two newspapers are estimating tomorrow’s chances for the Chinese shares. The first newspaper estimates the shares will go down as low as 8%. The second newspaper estimates that the Chinese shares will turnaround with an increase of 2%.
a. What approach of probability assessment is used by the newspapers?
b. How did the newspapers make their estimations?
c. Why did the two newspapers provide two different judgments by using the same probability assessment approach?
4-15. A school is going to conduct cocurricular activities for its students. The teachers conducted a survey to find out the students’ areas of interest and asked them to choose their favorite activity. From a sample of 100 students, the teachers found that 32 students are interested in music and dancing; 25 students like drawing; 28 students enjoy sports; and 15 students like gardening. Due to the available budget, the school can only provide two activities.
a. Which two activities will be their first choices?
b. What probability assessment approach is used in selecting the two activities?
c. Are the activities chosen by the students mutually exclusive? Explain.
4-16. One automobile dealership sells Buick, Cadillac, and Chevrolet automobiles. It also sells used cars that it gets as trade-ins on new car purchases. If two cars are sold on Tuesday by the dealership, what is the sample space for the type of cars that might be sold?
4-17. Golden Screen Cinema (GSC) is showcasing two popular Hollywood movies—Harry Potter and The Pirates of the Caribbean. Both movies are being shown in large auditoriums that can accommodate a maximum of 200 people. According to the number of tickets sold on a specific day, GSC found that 145 people watched Harry Potter and 156 audiences watched The Pirates of the Caribbean.
a. What probability assessment approach is suitable for GSC to judge the likelihood of the choice made by the people?
b. GSC is arranging for the next week’s movie schedule. Based on the information, which movie is going to have more show times?
4-18. A survey of 490 workers included the question “In your opinion, is it OK for a company to monitor its employees’ Internet use?” The possible responses were:
(1) Only after informing the employees, (2) Does not need to inform the employees, (3) Only when company
believes an employee is misusing the Internet, (4) Company does not have the right, and (5) Only if an employee has previously misused the Internet. The following table contains the results for the respondents:
Response 1 2 3 4 5
Number of Respondents 150 206 109 11 14 a. Calculate the probability that a randomly chosen
respondent would indicate that there should be some restriction concerning the company’s right to monitor Internet use.
b. Indicate the method of probability assessment used to determine the probability in part a.
c. Are the events that a randomly selected respondent chose response 1 and that another randomly selected respondent chose response 2 independent, mutually exclusive, or dependent events? Explain.
4-19. At Chrisdale Kindergarten, there are three classes for 148 students in their first year. Anna is one of the teachers at Chrisdale and she is interested to know about the students’ favorite pet. She conducted a survey and the results show that 65 students like dogs, 54 students like cats, 12 students like fish, and the remaining chose the “others” option provided in her survey form.
a. If a student is selected at random, what is the probability that they will not find their favorite pet listed in the survey?
b. Define the probability assessment approach used in determining the answer for part a.
c. Determine the sample space for selecting two students at random.
4-20. Two agents joined an insurance company a year ago.
During the year, the first agent sold 32 life insurance policies, with an insurance sum of $42 million. For the same type of policy, the second agent sold 29 with a sum of $46 million. The company’s sales report for the year state that 525 life insurance policies have been sold, totaling to $450 million.
a. If the agency’s manager wanted to determine the new agents’ sales probability, what probability assessment approach should he use?
b. Compute the probability of the number of life insurance policies sold by both new agents.
c. The manager needs each agent to achieve a probability of 10% sales in insured sum. Determine whether both the new agents fulfilled their
manager’s requirement.
Computer Software Exercises
4-21. The data in the file called Babies indicate whether the past 50 babies delivered at a local hospital were delivered using the caesarean method.
a. Based on these data, what is the probability that a baby born in this hospital will be born using the caesarean method?
b. What concerns might you have about using these data to assess the probability of a caesarean birth? Discuss.
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4.2 The Rules of Probability
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Chapter 4 1654-22. Several years ago, a large state university conducted a survey of undergraduate students regarding their use of computers. The results of the survey are contained in the data file ComputerUse.
a. Based on the data from the survey, what was the probability that undergraduate students at this university will have a major that requires them to use a computer on a daily basis?
b. Based on the data from this survey, if a student was a business major, what was the probability of the student believing that the computer lab facilities are very adequate?
c. Select a sample of 20 students at your university and ask them what their major is and whether the major requires them to access the Internet for computational purposes, with a personal computer, notebook, tablet, or other device. Use these data to calculate the probability of each major requiring a computing device, and then compare your results with those from the older survey.
4-23. 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 and the relative frequency assessment method to find the following probabilities:
a. What is the probability a scooter was assembled at the Tyler plant?
b. What is the probability that a scooter breakdown was due to a mechanical problem?
c. What is the probability that a scooter was assembled at the Lincoln plant and had an electrical problem?
4-24. A survey on cell phone use asked, in part, what was the most important reason 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 the 1,088 respondents.
a. Calculate the probability that a randomly chosen respondent would not use a wireless phone exclusively because of some type of difficulty in placing and receiving calls with a wireless phone.
b. Calculate the probability that a randomly chosen person would not use a wireless phone exclusively because of some type of difficulty in placing and receiving calls with a wireless phone and is over the age of 55.
c. Determine the probability that a randomly chosen person would not use a wireless phone exclusively because of a perceived need for Internet access and the safety of a traditional phone.
d. Of those respondents under 36, determine the probability that an individual in this age group would not use a wireless phone exclusively because of some type of difficulty in placing and receiving calls with a wireless phone.
4-25. A selection of soft-drink users is asked to taste two disguised soft drinks and indicate which they prefer.
The file titled Challenge contains the results of a study that was conducted on a college campus.
a. Determine the probability that a randomly chosen student prefers Pepsi.
b. Determine the probability that one of the students prefers Pepsi and is less than 20 years old.
c. Of those students who are less than 20 years old, calculate the probability that a randomly chosen student prefers (1) Pepsi and (2) Coke.
d. Of those students who are at least 20 years old, calculate the probability that a randomly chosen student prefers (1) Pepsi and (2) Coke.