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?

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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?

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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?

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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.

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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.

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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?

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A survey company asked purchasing managers what traits in a

A survey company asked purchasing managers what traits in a sales representative impressed them most. Seventy-eight per cent selected ‘thoroughness’. Forty per cent responded ‘knowledge of your own product’. The purchasing managers were allowed to list more than one trait. Suppose 27% of the purchasing managers listed both ‘thoroughness’ and ‘knowledge of your own product’ as the traits that impressed them most. A purchasing manager is sampled randomly.

(a) What is the probability that the manager selected ‘thoroughness’ or ‘knowledge of your own product’?

(b) What is the probability that the manager did not select ‘thoroughness’ and did not select ‘knowledge of your own product’?

(c) If it is known that the manager selected ‘thoroughness’, what is the probability that the manager also selected ‘knowledge of your own product’?

(d) What is the probability that the manager did not select ‘thoroughness’ but did select ‘knowledge of your own product’?

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A white goods manufacturer is sourcing parts for its air-conditioning

A white goods manufacturer is sourcing parts for its air-conditioning units. Management is currently putting together a timeline for the next stages of manufacturing. To do so, management considers that one set of parts relating to the motor are coming from Germany and one set of parts relating to the casing will come from the USA. Past experience suggests there is an 8% probability that the German parts will arrive late due to disruptions in shipping. The probability of parts arriving late from the USA is deemed to be 5%.

(a) What is the probability that parts from Germany will not be delayed?

(b) What is the probability that parts from the USA will not be delayed?

(c) Assume that the events in the two previous questions are independent. What is the probability that the parts from Germany will arrive late given that parts from the USA arrived late?

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Octopus Travel studied the types of work-related activity that Australians

Octopus Travel studied the types of work-related activity that Australians do while on holiday. Among other things, 20% take their work with them on holiday. Fifteen per cent negotiate new deals or contracts on holiday. Respondents were allowed to select more than one activity. Suppose that, of those who take their work on holiday, 68% negotiate new deals or contracts. One of these survey respondents is selected randomly.

(a) What is the probability that, while on holiday, this respondent negotiated a new deal or contract and took their work?

(b) What is the probability that, while on holiday, this respondent did not take their work and did not negotiate a new deal or contract?

(c) What is the probability that, while on holiday, this respondent took their work given that the respondent negotiated a new deal or contract?

(d) What is the probability that, while on holiday, this respondent did not negotiate a new deal or contract given that the respondent took their work?

(e) What is the probability that, while on holiday, this respondent did not negotiate a new deal or contract given that the respondent did not take their work?

(f) Construct a probability matrix for this problem.

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A study of tweeting behaviour revealed that, among 36- to

A study of tweeting behaviour revealed that, among 36- to 45-year-olds, the number one tweeted topic relates to family, with 28% making some reference to it among those tweets that could be categorised in a mutually exclusive manner. Other tweets were categorised as referring to the arts (22%), entertainment (18%), education (17%) and sports (11%), while 4% referred to some other topic. If a tweet by a 36- to 45-year-old is randomly selected and able to be categorised in the same way, determine the probabilities of the following.

(a) The tweet is about family.

(b) The tweet is not about sports.

(c) The tweet is about the arts or entertainment.

(d) The tweet is neither about family nor about education.

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