## According to a recent survey, 40% of millennials (those born

According to a recent survey, 40% of millennials (those born in the 1980s or 1990s) view themselves more as ‘spenders’ than ‘savers’. The survey also reveals that 75% of millennials view social networking as important to find out about products and services. A social media expert wants to determine the probability that a randomly selected millennial either views themselves as a ‘spender’ or views social networking as important to find out about products and services. Can this question be answered? Under what conditions can it be solved? If the problem cannot be solved, what information is needed to make it solvable?

## According to a consumer report, approximately 3% of New Zealanders

According to a consumer report, approximately 3% of New Zealanders bought a new vehicle in the past 12 months. The report also indicates that 10% commenced a new job in the same period. The report further reveals that in the past 12 months, 2% bought a new vehicle and commenced a new job. A New Zealander is randomly selected.

(a) What is the probability that in the past 12 months the individual has purchased a new vehicle or commenced a new job?

(b) What is the probability that in the past 12 months the individual has purchased a new vehicle or commenced a new job but not both?

(c) What is the probability that in the past 12 months the individual has neither bought a new vehicle nor commenced a new job?

(d) Why does the special law of addition not apply to this problem?

## The operations manager of a cinema complex is interested in

The operations manager of a cinema complex is interested in how patrons using the candy bar purchase food and drinks in various combinations. In particular, the manager wants to know the probability that a patron will purchase water and popcorn, juice and popcorn, soft drink and popcorn, and popcorn with no drink. The table lists a random sample of 730 purchase outcomes recently made by patrons. To simplify the experiment, only outcomes in which no beverage or one beverage was purchased are examined. Hence, any event in any row is mutually exclusive of any event in any other row (e.g. a single purchase could be listed as purchasing water only, but outcomes relating to a single purchase involving water and juice are not recorded). Transform this table to list probability information of purchases and use it to respond to the manager’s questions. Overall, what is the probability that popcorn will be purchased?

## The two most popular majors in a business degree are

The two most popular majors in a business degree are marketing and international business, with 30% of students enrolled in marketing and 18% enrolled in international business. Suppose 20 students are selected at random.

(a) What is the probability that:

(i) at least half of them are studying a marketing major

(ii) no more than a quarter are studying an international business major

(iii) between 10 and 15 are studying a marketing major?

(b) Find the mean and variance of the number of students studying a marketing major. Find the mean and variance of the total number studying a marketing or international business major.

## A university conducted a study into students’ perceived usability of

A university conducted a study into students’ perceived usability of a proposed upgrade to the existing learning management system (LMS). Students were asked to visit a test site and evaluate the proposed LMS relative to their perceptions of the existing LMS. To further evaluate the results, participants were asked to nominate one field that represented their primary area of study. Some 84% of students perceived the proposed LMS to be superior in usability to the existing system. Of the students sampled, 45% were primarily studying business and rated the proposed system to be superior in usability. Of those students primarily studying science, 95% found the proposed LMS easier to use; these students represented 30% of the surveyed participants. Together, those students primarily studying science and those primarily studying business made up 85% of the sample. If a student is selected randomly, determine the probabilities of the following.

(a) The student rates the proposed system as superior in usability given that the student primarily studies business.

(b) The student rates the proposed system as superior in usability given that the student primarily studies in an area other than business.

(c) The student is studying science, given they do not believe the proposed system to be superior in usability.

(d) The student is primarily studying science and believes the proposed system to be superior in usability.

## The ABS energy survey reports that 45% of all Australian

The ABS energy survey reports that 45% of all Australian households have an air conditioner and that 30% of all Australian households have a dishwasher. An Australian household is randomly selected.

(a) Assume that whether a household has a dishwasher is unrelated to whether the same household has an air conditioner. Use the special law of multiplication to determine the probability that the household has both an air conditioner and a dishwasher.

(b) Suppose another report states that if an Australian household has a dishwasher, the probability of this household having an air conditioner is 80%. Use the general law of multiplication to determine the probability that the household has both an air conditioner and a dishwasher. Does it appear that whether a household has a dishwasher is related to whether the same household has an air conditioner?

## Many organisations, including the Cancer Society of New Zealand, endorse

Many organisations, including the Cancer Society of New Zealand, endorse the recommendation that people consume three servings of vegetables and two servings of fruit per day as this provides some protection from heart disease and cancer. A report states that 49% of New Zealanders regularly eat fresh fruit and vegetables and that, among this same group, 80% agree it is good for their health. Interestingly, in addition the report states that 94% of those who do not eat fresh fruit and vegetables regularly also believe eating fresh fruit and vegetables regularly is good for health. If a New Zealander is selected randomly, determine the probability that this person:

(a) regularly eats fresh fruit and vegetables, and believes this to be a dietary practice that is good for their health.

(b) does not regularly eat fresh fruit and vegetables.

(c) does not regularly eat fresh fruit and vegetables, yet believes this to be a dietary practice that is good for health.

(d) believes regularly eating fresh fruit and vegetables is a dietary practice that is good for health.

(e) regularly eats fresh fruit and vegetables, given they believe this to be a dietary practice that is good for their health.

## A company has recently undertaken a program to replace the

A company has recently undertaken a program to replace the desktop computers of its staff. The new computers have a different operating system and newer software. Initially, employees appeared disgruntled about having to learn how to use the computers. The company offered, at great cost, a series of training programs; unfortunately, some employees still appear to be unsatisfied, although management suspects this stems from part-time employees who have had fewer opportunities to use the new computers.

Assess a survey that was conducted on a random sample of 200 employees to report on the interest in training overall, but also determine whether the additional training sought by employees is conditional on their employment status being full-time or part-time. Write a short summary to help management decide whether to offer more training and whether this arises from the part-time cohort.

## In any given month, a busy port can have many

In any given month, a busy port can have many cargo ships arriving. The following table displays the data collected over a month of ship arrivals at a particular port, with a total of 76. This data will allow the manager of the port to formulate staffing plans for crews to load and unload the vessels.

Assume that the number of arrivals per day has a Poisson distribution.

(a) What is the probability of zero arrivals on any given day?

(b) What is the probability of two arrivals on any given day?

(c) The manager’s standard plan should provide a 90% service rate — it should include adequate labour and other resources to service 90% of the vessels on their arrival date. How many arrivals per day should the standard plan anticipate?