2.04b Binomial distribution: as model B(n,p)

The queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function $$\text{f}(x) = \begin{cases} kx(81 - x^2) & 0 \leqslant x \leqslant 9 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that \(k = \frac{4}{6561}\). [3]

Using integration, find

1. the mean queuing time of a customer, [4]
2. the probability that a customer will queue for more than 5 minutes. [3]

Three independent customers shop at the post office.

1. Find the probability that at least 2 of the customers queue for more than 5 minutes. [3]

Past records show that 20\% of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps was taken and 2 of them had bought them in single packets.

1. Use these data to test, at the 5\% level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly. [6]

At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03. To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.

1. Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. Write down the significance level of this test. [1]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm{N}(\mu, 0.3^2)\).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

In a town, 30\% of residents listen to the local radio station. Four residents are chosen at random.

1. State the distribution of the random variable \(X\), the number of these four residents that listen to local radio. [2]
2. On graph paper, draw the probability distribution of \(X\). [3]
3. Write down the most likely number of these four residents that listen to the local radio station. [1]
4. Find E(\(X\)) and Var (\(X\)). [3]

1. Write down the conditions under which the binomial distribution may be a suitable model to use in statistical work. [4]

A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that

1. 1. the first 5 will occur on the sixth throw,
   2. in the first eight throws there will be exactly three 5s.
   [8]

1. State two conditions under which a random variable can be modelled by a binomial distribution. [2]

In the production of a certain electronic component it is found that 10% are defective. The component is produced in batches of 20.

1. Write down a suitable model for the distribution of defective components in a batch. [1]

Find the probability that a batch contains

1. no defective components, [2]
2. more than 6 defective components. [2]
3. Find the mean and the variance of the defective components in a batch. [2]

A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.

1. Using a suitable approximation, find the probability that the supplier will receive a refund. [4]

A manufacturer produces large quantities of coloured mugs. It is known from previous records that 6\% of the production will be green. A random sample of 10 mugs was taken from the production line.

1. Define a suitable distribution to model the number of green mugs in this sample. [1]
2. Find the probability that there were exactly 3 green mugs in the sample. [3]

A random sample of 125 mugs was taken.

1. Find the probability that there were between 10 and 13 (inclusive) green mugs in this sample, using
   1. a Poisson approximation, [3]
   2. a Normal approximation. [6]

Bhim and Joe play each other at badminton and for each game, independently of all others, the probability that Bhim loses is 0.2 Find the probability that, in 9 games, Bhim loses

1. exactly 3 of the games, [3]
2. fewer than half of the games. [2]

Bhim attends coaching sessions for 2 months. After completing the coaching, the probability that he loses each game, independently of all others, is 0.05 Bhim and Joe agree to play a further 60 games.

1. Calculate the mean and variance for the number of these 60 games that Bhim loses. [2]
2. Using a suitable approximation calculate the probability that Bhim loses more than 4 games. [3]

A company claims that a quarter of the bolts sent to them are faulty. To test this claim the number of faulty bolts in a random sample of 50 is recorded.

1. Give two reasons why a binomial distribution may be a suitable model for the number of faulty bolts in the sample. [2]
2. Using a 5\% significance level, find the critical region for a two-tailed test of the hypothesis that the probability of a bolt being faulty is \(\frac{1}{4}\). The probability of rejection in either tail should be as close as possible to 0.025 [3]
3. Find the actual significance level of this test. [2]

In the sample of 50 the actual number of faulty bolts was 8.

1. Comment on the company's claim in the light of this value. Justify your answer. [2]

The machine making the bolts was reset and another sample of 50 bolts was taken. Only 5 were found to be faulty.

1. Test at the 1\% level of significance whether or not the probability of a faulty bolt has decreased. State your hypotheses clearly. [6]

The proportion of houses in Radville which are unable to receive digital radio is 25\%. In a survey of a random sample of 30 houses taken from Radville, the number, \(X\), of houses which are unable to receive digital radio is recorded.

1. Find P(5 \(\leq X < 11\)) [3]

A radio company claims that a new transmitter set up in Radville will reduce the proportion of houses which are unable to receive digital radio. After the new transmitter has been set up, a random sample of 15 houses is taken, of which 1 house is unable to receive digital radio.

1. Test, at the 10\% level of significance, the radio company's claim. State your hypotheses clearly. [5]

A manufacturer of chocolates produces 3 times as many soft centred chocolates as hard centred ones. Assuming that chocolates are randomly distributed within boxes of chocolates, find the probability that in a box containing 20 chocolates there are

1. equal numbers of soft centred and hard centred chocolates, [3]
2. fewer than 5 hard centred chocolates. [2]

A large box of chocolates contains 100 chocolates.

1. Write down the expected number of hard centred chocolates in a large box. [2]

In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.

1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]

Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.

1. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. State the significance level of this test giving your answer to 2 significant figures. [1]

Data were collected on the number of female puppies born in 200 litters of size 8. It was decided to test whether or not a binomial model with parameters \(n = 8\) and \(p = 0.5\) is a suitable model for these data. The following table shows the observed frequencies and the expected frequencies, to 2 decimal places, obtained in order to carry out this test.

1. Find the values of \(R\), \(S\) and \(T\). [4]
2. Carry out the test to determine whether or not this binomial model is a suitable one. State your hypotheses clearly and use a 5\% level of significance. [7]

An alternative test might have involved estimating \(p\) rather than assuming \(p = 0.5\).

1. Explain how this would have affected the test. [1]

Data were collected on the number of female puppies born in 200 litters of size 8. It was decided to test whether or not a binomial model with parameters \(n = 8\) and \(p = 0.5\) is a suitable model for these data. The following table shows the observed frequencies and the expected frequencies, to 2 decimal places, obtained in order to carry out this test.

1. Find the values of \(R\), \(S\) and \(T\). [4]
2. Carry out the test to determine whether or not this binomial model is a suitable one. State your hypotheses clearly and use a 5\% level of significance. [7]

An alternative test might have involved estimating \(p\) rather than assuming \(p = 0.5\).

1. Explain how this would have affected the test. [1]

Among the families with two children in a large city, the probability that the elder child is a boy is \(\frac{5}{12}\) and the probability that the younger child is a boy is \(\frac{9}{16}\). The probability that the younger child is a girl, given that the elder child is a girl, is \(\frac{1}{4}\). One of the families is chosen at random. Using a tree diagram, or otherwise,

1. show that the probability that both children are boys is \(\frac{1}{8}\). [5 marks]

Find the probability that

1. one child is a boy and the other is a girl, [3 marks]
2. one child is a boy given that the other is a girl. [3 marks]

If three of the families are chosen at random,

1. find the probability that exactly two of the families have two boys. [3 marks]
2. State an assumption that you have made in answering part (d). [1 mark]

The students in a large Sixth Form can choose to do exactly one of Community Service, Games or Private Study on Wednesday afternoons. The probabilities that a randomly chosen student does Games and Private Study are \(\frac{3}{8}\) and \(\frac{1}{5}\) respectively. It may be assumed that the number of students is large enough for these probabilities to be treated as constant.

1. Find the probability that a randomly chosen student does Community Service. [2 marks]
2. If two students are chosen at random, find the probability that they both do the same activity. [3 marks]
3. If three students are chosen at random, find the probability that exactly one of them does Games. [3 marks]

Two-fifths of the students are girls, and a quarter of these girls do Private Study.

1. Find the probability that a randomly chosen student who does Private Study is a boy. [5 marks]

A certain four-sided die is biased. The score, \(X\), on each throw is a random variable with probability distribution as shown in the table. Throws of the die are independent.

\(x\)	0	1	2	3
P\((X = x)\)	\(\frac{1}{2}\)	\(\frac{1}{4}\)	\(\frac{1}{8}\)	\(\frac{1}{8}\)

1. Calculate E\((X)\) and Var\((X)\). [5]

The die is thrown 10 times.

1. Find the probability that there are not more than 4 throws on which the score is 1. [2]
2. Find the probability that there are exactly 4 throws on which the score is 2. [3]

The probability distribution of a random variable \(X\) is shown.

\(x\)	1	3	5	7
P\((X = x)\)	0.4	0.3	0.2	0.1

1. Find E\((X)\) and Var\((X)\). [5]
2. Three independent values of \(X\), denoted by \(X_1\), \(X_2\) and \(X_3\), are chosen. Given that \(X_1 + X_2 + X_3 = 19\), write down all the possible sets of values for \(X_1\), \(X_2\) and \(X_3\) and hence find P\((X_1 = 7)\). [2]
3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5s. [3]

In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows. • If \(X < 2\), the batch is accepted. • If \(X > 2\), the batch is rejected. • If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted. It is given that 5\% of mugs are defective.

1. 1. Find the probability that the batch is rejected after just the first sample is checked. [3]
   2. Show that the probability that the batch is rejected is 0.327, correct to 3 significant figures. [5]
2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked. [3]

In a multiple-choice test there are 30 questions. For each question, there is a 60% chance that a randomly selected student answers correctly, independently of all other questions.

1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct. [3]
2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct. [2]

An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.

1. Three days are chosen at random. Find the probability that the pollution level is
   1. low on all 3 days, [2]
   2. low on at least one day, [2]
   3. low on one day, medium on another day, and high on the other day. [3]
2. Ten days are chosen at random. Find the probability that
   1. there are no days when the pollution level is high, [2]
   2. there is exactly one day when the pollution level is high. [3]

The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.

1. Carry out a test at the 5% level to determine if there is evidence to suggest that she is correct. Use hypotheses \(H_0: p = 0.5\), \(H_1: p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(H_1\) has this form. [5]

Mark is playing solitaire on his computer. The probability that he wins a game is 0.2, independently of all other games that he plays.

1. Find the expected number of wins in 12 games. [2]
2. Find the probability that
   1. he wins exactly 2 out of the next 12 games that he plays, [3]
   2. he wins at least 2 out of the next 12 games that he plays. [3]
3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the 5\% level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test. [9]

25% of the plants of a particular species have red flowers. A random sample of 6 plants is selected.

1. Find the probability that there are no plants with red flowers in the sample. [2]
2. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers. [2]

Any patient who fails to turn up for an outpatient appointment at a hospital is described as a 'no-show'. At a particular hospital, on average 15% of patients are no-shows. A random sample of 20 patients who have outpatient appointments is selected.

1. Find the probability that
   1. there is exactly 1 no-show in the sample, [3]
   2. there are at least 2 no-shows in the sample. [2]

The hospital management introduces a policy of telephoning patients before appointments. It is hoped that this will reduce the proportion of no-shows. In order to check this, a random sample of \(n\) patients is selected. The number of no-shows in the sample is recorded and a hypothesis test is carried out at the 5% level.

1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
2. In the case that \(n = 20\) and the number of no-shows in the sample is 1, carry out the test. [4]
3. In another case, where \(n\) is large, the number of no-shows in the sample is 6 and the critical value for the test is 8. Complete the test. [3]
4. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the 5% level. [2]