Review practice · Section 3 of 3

One question in a survey is as follows:

“Do you use Facebook during office hours at least once a day? Yes or No?”

It is known that on average, 7 out of 9 staff members answered “yes” to the above question.

For a random sample of 60 staff members,

(i) find the standard deviation of the number of staff members who answer “yes”,1 mark

(ii) find the probability that the number of staff members who answer “yes” exceeds the mean,3 marks

(iii) find the most probable number of staff members who answer “yes”,2 marks

(iv) state, with a reason, if the above answers are valid if the total number of staff members in the company is small.1 mark

The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$.

Denoting $P(X=k)$ by $p_k$, it is given that $\dfrac{p_k}{p_{k-1}}=\dfrac{(n-k+1)p}{k(1-p)}$, $k=1,2,\ldots,n$.

A distribution is said to be bimodal if it has two modes.

Find the least value of $n$, and the corresponding modes of $X$, given that $X$ is bimodal and that $p=\dfrac{18}{25}$.3 marks

A jar contains 10 blue and 8 red marbles. Five marbles are randomly drawn from the box, one by one and without replacement.

(i) Explain why it is inappropriate to model the number of blue marbles by a binomial distribution.1 mark

(ii) Find the probability that exactly three marbles are blue.3 marks

Another jar contains 20 blue and 12 red marbles. $n$ marbles are randomly drawn from the box, one by one and with replacement. The number of red marbles drawn is denoted by $R$.

(iii) Given that the mean of $R$ is 4.5, find $n$ and $P(R>4)$.3 marks

(iv) Given instead that $P(R=0\text{ or }1)<0.01$, write down an inequality for $n$ and find the least value of $n$.3 marks

A firm sells two types of electrical components, $A$ and $B$.

Type $A$ components are packed in boxes of 50. On average, 2% of the Type $A$ components are faulty.

(i) State, in context, two assumptions needed for the number of faulty components of Type $A$ in a box to be well-modelled by a binomial distribution.2 marks

You are now given that the number of faulty components of type $A$ in a box follows a binomial distribution.

(ii) Find the probability that a box of these components contains more than 1 faulty component.2 marks

Five boxes of Type $A$ components are selected at random.

(iii) Find the probability that the fifth box is the third box selected which has more than one faulty component.2 marks

(iv) Find the probability that there are 3 boxes with more than 1 faulty component each.2 marks

(v) Explain why the answer to part (iv) is greater than the answer to part (iii).1 mark

During a pandemic, an unknown infectious disease spread through a population. Researchers wanted to study the probability, $p$, that a person is infected. They collected some data from a random sample of 25 people.

(i) State, in the context of this question, two assumptions needed for the number of infected people to be well modelled by a binomial distribution.2 marks

Assume now that these assumptions do in fact hold.

(ii) It is given that the probability that at most 2 people are infected is 0.0982. Write an equation in $p$ and solve for $p$.2 marks

A total of 30 random samples of 25 people were chosen from this population.

(iii) Find the probability that the total number of samples with at most 2 infected people is at least 5.3 marks

It is now given that $p=0.10$.

A diagnostic test for the presence of the disease was carried out on the population. It was found that the percentage of infected people who were tested positive is 85%, while the percentage of non-infected people who were tested negative is 96%.

(iv) Find the probability that a person who tested positive carries the disease.3 marks

(v) Hence or otherwise, find the probability that a person who tested positive does not carry the disease.1 mark

(vi) Discuss briefly if the diagnostic test is worthwhile.1 mark

An airline company sold 105 tickets for a particular flight. Based on past experience, 8% of passengers do not turn up for the flight. Let $X$ be the number of passengers who do not turn up for that flight.

(i) State, in context, two assumptions needed for $X$ to be well modelled by a binomial distribution.2 marks

(ii) Find the probability that fewer than 5 do not turn up for the flight.2 marks

“Overbooking” occurs when there are more passengers who turn up than the number of seats available for the flight. When this happens, the airline will re-accommodate the excess passengers, that is, arrangements will be made for them to take another flight.

(iii) It is given that the particular flight has only 100 seats. Find the probability that at least 3 passengers have to be re-accommodated given that “overbooking” has occurred.3 marks