Methods2024 NHT Exam 2 Section B · Q3CAS active12 marks

Office attendance, electricity costs and defective LED sensors

A small business has four people working full-time. The business rents an office, and on most days some people work from home.

On any given workday, the number of people working in the office can be modelled by the discrete random variable, $N$ , with the following probability distribution.

| $n$ | $1$ | $2$ | $3$ | $4$ | |---|---|---|---|---| | | | | | |

Find $\operatorname{E}(N)$ , the expected value for the number of people working in the office.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Find the probability that all four people are working in the office, given that at least two people are working in the office.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

In a fortnight with ten workdays, find the probability that all four people are working in the office on at least three days.

Write your answer correct to four decimal places.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

d.i.

The business owner is trying to find ways to reduce costs.

Her daily electricity costs, in dollars, can be modelled using a continuous random variable, $C$ , which has the probability density function

$f(c) = \begin{cases} \dfrac{3c(c-10)^2}{2500} & 0 \le c \le 10 \\ 0 & \text{elsewhere} \end{cases}$

Identify the maximum daily electricity cost, in dollars, for this small business.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

d.ii.

Find the mean daily electricity cost, in dollars.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

d.iii.

Find the probability that, on a given day, the electricity cost is within two standard deviations of the mean.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

Methods2024 NHT Exam 2 Section B · Q3CAS active12 marks

Office attendance, electricity costs and defective LED sensors

A small business has four people working full-time. The business rents an office, and on most days some people work from home.

On any given workday, the number of people working in the office can be modelled by the discrete random variable, $N$ , with the following probability distribution.

| $n$ | $1$ | $2$ | $3$ | $4$ | |---|---|---|---|---| | | | | | |

Find $\operatorname{E}(N)$ , the expected value for the number of people working in the office.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Find the probability that all four people are working in the office, given that at least two people are working in the office.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

In a fortnight with ten workdays, find the probability that all four people are working in the office on at least three days.

Write your answer correct to four decimal places.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

d.i.

The business owner is trying to find ways to reduce costs.

Her daily electricity costs, in dollars, can be modelled using a continuous random variable, $C$ , which has the probability density function

$f(c) = \begin{cases} \dfrac{3c(c-10)^2}{2500} & 0 \le c \le 10 \\ 0 & \text{elsewhere} \end{cases}$

Identify the maximum daily electricity cost, in dollars, for this small business.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

d.ii.

Find the mean daily electricity cost, in dollars.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

d.iii.

Find the probability that, on a given day, the electricity cost is within two standard deviations of the mean.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.