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

2 Every evening, 5 men and 5 women are chosen to take part in a phone-in competition. Of these 10 people, exactly 3 will win a prize. These 3 prize-winners are chosen at random.

1. Find the probability that, on a particular evening, 2 of the prize-winners are women and the other is a man. Give your answer as a fraction in its simplest form.
2. Four evenings are selected at random. Find the probability that, on at least three of the four evenings, 2 of the prize-winners are women and the other is a man.

3 The weights of bags of a particular brand of flour are quoted as 1.5 kg . In fact, on average \(10 \%\) of bags are underweight.

1. Find the probability that, in a random sample of 50 bags, there are exactly 5 bags which are underweight.
2. Bags are randomly chosen and packed into boxes of 20 . Find the probability that there is at least one underweight bag in a box.
3. A crate contains 48 boxes. Find the expected number of boxes in the crate which contain at least one underweight bag.

4 Martin has won a competition. For his prize he is given six sealed envelopes, of which he is allowed to open exactly three and keep their contents. Three of the envelopes each contain \(\pounds 5\) and the other three each contain \(\pounds 1000\). Since the envelopes are identical on the outside, he chooses three of them at random. Let \(\pounds X\) be the total amount of money that he receives in prize money.

1. Show that \(\mathrm { P } ( X = 15 ) = 0.05\). The probability distribution of \(X\) is given in the table below.

   \(r\)	15	1010	2005	3000
   \(\mathrm { P } ( X = r )\)	0.05	0.45	0.45	0.05

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 A normal pack of 52 playing cards contains 4 aces. A card is drawn at random from the pack. It is then replaced and the pack is shuffled, after which another card is drawn at random.

1. Find the probability that neither card is an ace.
2. This process is repeated 10 times. Find the expected number of times for which neither card is an ace.

7 A drug for treating a particular minor illness cures, on average, \(78 \%\) of patients. Twenty people with this minor illness are selected at random and treated with the drug.

1. \(( A )\) Find the probability that exactly 19 patients are cured.
   (B) Find the probability that at most 18 patients are cured. \(( C )\) Find the expected number of patients who are cured.
2. A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the \(1 \%\) significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
3. If the researchers had chosen to carry out the hypothesis test at the \(5 \%\) significance level, what would the result have been? Justify your answer.

8 A sales office employs 21 representatives. Each day, for each representative, the probability that he or she achieves a sale is 0.7 , independently of other representatives. The total number of representatives who achieve a sale on any one day is denoted by \(K\).

1. Using a suitable approximation (which should be justified), find \(\mathrm { P } ( K \geqslant 16 )\).
2. Using a suitable approximation (which should be justified), find the probability that the mean of 36 observations of \(K\) is less than or equal to 14.0 . 4

3 The probability that a randomly chosen PPhone has a faulty casing is 0.0228 . A random sample of 200 PPhones is obtained. Use a suitable approximation to find the probability that the number of PPhones in the sample with a faulty casing is 2 or fewer. Justify your approximation.

8 A company has 3600 employees, of whom \(22.5 \%\) live more than 30 miles from their workplace. A random sample of 40 employees is obtained.

1. Use a suitable approximation, which should be justified, to find the probability that more than 5 of the employees in the sample live more than 30 miles from their workplace.
2. Describe how to use random numbers to select a sample of 40 from a population of 3600 employees.

9 A pharmaceutical company is developing a new drug to treat a certain disease. The company will continue to develop the drug if the proportion \(p\) of those who have the disease and show a substantial improvement after treatment is greater than 0.7 . The company carries out a test, at the \(5 \%\) significance level, on a random sample of 14 patients who suffer from the disease.

1. Find the critical region for the test.
2. Given that 12 of the 14 patients in the sample show a substantial improvement, carry out the test.
3. Find the probability that the test results in a Type II error if in fact \(p = 0.8\). RECOGNISING ACHIEVEMENT

3 An electronics company is developing a new sound system. The company claims that \(60 \%\) of potential buyers think that the system would be good value for money. In a random sample of 12 potential buyers, 4 thought that it would be good value for money. Test, at the 5\% significance level, whether the proportion claimed by the company is too high.

2 A university has a large number of students, of whom \(35 \%\) are studying science subjects. A sample of 10 students is obtained by listing all the students, giving each a serial number and selecting by using random numbers.

1. Find the probability that fewer than 3 of the sample are studying science subjects.
2. It is required that, in selecting the sample, the same student is not selected twice. Explain whether this requirement invalidates your calculation in part (i).

8 The random variable \(R\) has the distribution \(\mathrm { B } ( 14 , p )\). A test is carried out at the \(\alpha \%\) significance level of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\), against \(\mathrm { H } _ { 1 } : p > 0.25\).

1. Given that \(\alpha\) is as close to 5 as possible, find the probability of a Type II error when the true value of \(p\) is 0.4 .
2. State what happens to the probability of a Type II error as
   1. \(p\) increases from 0.4,
   2. \(\alpha\) increases, giving a reason.

2 Clover stems usually have three leaves. Occasionally a clover stem has four leaves. This is considered by some to be lucky and is known as a four-leaf clover. On average 1 in 10000 clover stems is a four-leaf clover. You may assume that four-leaf clovers occur randomly and independently. A random sample of 5000 clover stems is selected.

1. State the exact distribution of \(X\), the number of four-leaf clovers in the sample.
2. Explain why \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Use this Poisson distribution to find the probability that the sample contains at least one four-leaf clover.
4. Find the probability that in 20 samples, each of 5000 clover stems, there are exactly 9 samples which contain at least one four-leaf clover.
5. Find the expected number of these 20 samples which contain at least one four-leaf clover. The table shows the numbers of four-leaf clovers in these 20 samples.

   Number of four-leaf clovers	0	1	2	\(> 2\)
   Number of samples	11	7	2	0

6. Calculate the mean and variance of the data in the table.
7. Briefly comment on whether your answers to parts (v) and (vi) support the use of the Poisson approximating distribution in part (iii).

2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.

1. Find the probability that exactly 1 of a batch of 10 laptops is faulty.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
   (A) there are no faulty laptops in the batch,
   (B) there are more than the expected number of faulty laptops in the batch.
4. A large company buys a batch of 2000 of these laptops for its staff.
   (A) State the exact distribution of the number of faulty laptops in this batch.
   (B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.

2 A particular genetic mutation occurs in one in every 300 births on average. A random sample of 1200 births is selected.

1. State the exact distribution of \(X\), the number of births in the sample which have the mutation.
2. Explain why \(X\) has, approximately, a Poisson distribution.
3. Use a Poisson approximating distribution to find
   (A) \(\mathrm { P } ( X = 1 )\),
   (B) \(\mathrm { P } ( X > 4 )\).
4. Twenty independent samples, each of 1200 births, are selected. State the mean and variance of a Normal approximating distribution suitable for modelling the total number of births with the mutation in the twenty samples.
5. Use this Normal approximating distribution to
   (A) find the probability that there are at least 90 births which have the mutation,
   ( \(B\) ) find the least value of \(k\) such that the probability that there are at most \(k\) births with this mutation is greater than 5\%.

2 Suppose that 3\% of the population of a large city have red hair.

1. A random sample of 10 people from the city is selected. Find the probability that there is at least one person with red hair in this sample. A random sample of 60 people from the city is selected. The random variable \(X\) represents the number of people in this sample who have red hair.
2. Explain why the distribution of \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Hence find
   (A) \(\mathrm { P } ( X = 2 )\),
   (B) \(\mathrm { P } ( X > 2 )\).
4. Discuss whether or not it would be appropriate to model \(X\) using a Normal approximating distribution. A random sample of 5000 people from the city is selected.
5. State the exact distribution of the number of people with red hair in the sample.
6. Use a suitable Normal approximating distribution to find the probability that there are at least 160 people with red hair in the sample.

3 The manufacturer of electronic components uses the following process to test the proportion of defective items produced. A random sample of 20 is taken from a large batch of components.

• If no defective item is found, the batch is accepted.
• If two or more defective items are found, the batch is rejected.
• If one defective item is found, a second random sample of 20 is taken. If two or more defective items are found in this second sample, the batch is rejected, otherwise the batch is accepted.

The proportion of defective items in the batch is denoted by \(p\), and \(q = 1 - p\).

1. Show that the probability that a batch is accepted is \(q ^ { 20 } + 20 p q ^ { 38 } ( q + 20 p )\). For a particular component, \(p = 0.01\).
2. Given that a batch is accepted, find the probability that it is accepted as a result of the first sample.

Each of 200 identically biased dice is thrown repeatedly until an even number is obtained. The number of throws, \(x\), needed is recorded and the results are summarised in the following table. State a type of distribution that could be used to fit the data given in the table above. Fit a distribution of this type in which the probability of throwing an even number for each die is 0.6 and carry out a goodness of fit test at the 5\% significance level. For each of these dice, it is known that the probability of obtaining a 6 when it is thrown is 0.25 . Ten of these dice are each thrown 5 times. Find the probability that at least one 6 is obtained on exactly 4 of the 10 dice.

8 Drinking glasses are sold in packs of 4. The manufacturer conducts a survey to assess the quality of the glasses. The results from a sample of 50 randomly chosen packs are summarised in the following table. Fit a binomial distribution to the data and carry out a goodness of fit test at the \(10 \%\) significance level.

4 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.

1. Find the probability that in a batch of 50 there is
   (A) exactly one faulty teapot,
   (B) more than one faulty teapot.
2. The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.

7 An online shopping company takes orders through its website. On average \(80 \%\) of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.

1. Find the probability that
   (A) exactly 8 of these orders are delivered within 24 hours,
   (B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than \(80 \%\) of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
2. Write down suitable hypotheses and carry out a test at the \(5 \%\) significance level to determine whether there is any evidence to support the quality controller's suspicion.
3. A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the \(5 \%\) significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.

7 To withdraw money from a cash machine, the user has to enter a 4-digit PIN (personal identification number). There are several thousand possible 4-digit PINs, but a survey found that \(10 \%\) of cash machine users use the PIN '1234'.

1. 16 cash machine users are selected at random.
   (A) Find the probability that exactly 3 of them use 1234 as their PIN.
   (B) Find the probability that at least 3 of them use 1234 as their PIN.
   (C) Find the expected number of them who use 1234 as their PIN. An advertising campaign aims to reduce the number of people who use 1234 as their PIN. A hypothesis test is to be carried out to investigate whether the advertising campaign has been successful.
2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
3. A random sample of 20 cash machine users is selected.
   (A) Explain why the test could not be carried out at the \(10 \%\) significance level.
   (B) The test is to be carried out at the \(k \%\) significance level. State the lowest integer value of \(k\) for which the test could result in the rejection of the null hypothesis.
4. A new random sample of 60 cash machine users is selected. It is found that 2 of them use 1234 as their PIN. You are given that, if \(X \sim \mathrm {~B} ( 60,0.1 )\), then (to 4 decimal places) $$\mathrm { P } ( X = 2 ) = 0.0393 , \quad \mathrm { P } ( X < 2 ) = 0.0138 , \quad \mathrm { P } ( X \leqslant 2 ) = 0.0530 .$$ Using the same hypotheses as in part (ii), carry out the test at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}

9 Briony suspects that a particular 6-sided dice is biased in favour of 2. She plans to throw the dice 35 times and note the number of times that it shows a 2 . She will then carry out a test at the \(4 \%\) significance level. Find the rejection region for the test.

13 It is known that \(26 \%\) of adults in the UK use a certain app. A researcher selects a random sample of 5000 adults in the UK. The random variable \(X\) is defined as the number of adults in the sample who use the app. Given that \(\mathrm { P } ( X < n ) < 0.025\), calculate the largest possible value of \(n\).

14 The probability distribution of a random variable \(X\) is modelled as follows. \(\mathrm { P } ( X = x ) = \begin{cases} \frac { k } { x } & x = 1,2,3,4 , \\ 0 & \text { otherwise, } \end{cases}\) where \(k\) is a constant.

1. Show that \(k = \frac { 12 } { 25 }\).
2. Show in a table the values of \(X\) and their probabilities.
3. The values of three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\). Find \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } + X _ { 3 } \right)\). In a game, a player notes the values of successive independent observations of \(X\) and keeps a running total. The aim of the game is to reach a total of exactly 7 .
4. Determine the probability that a total of exactly 7 is first reached on the 5th observation. \section*{OCR} Oxford Cambridge and RSA