2.04c Calculate binomial probabilities

1. A large number of students sat an examination. All of the students answered the first question. The first question was answered correctly by \(40 \%\) of the students.

In a random sample of 20 students who sat the examination, \(X\) denotes the number of students who answered the first question correctly.

1. Write down the distribution of the random variable \(X\)
2. Find \(\mathrm { P } ( 4 \leqslant X < 9 )\) Students gain 7 points if they answer the first question correctly and they lose 3 points if they do not answer it correctly.
3. Find the probability that the total number of points scored on the first question by the 20 students is more than 0
4. Calculate the variance of the total number of points scored on the first question by a random sample of 20 students.

1. A shop sells rods of nominal length 200 cm . The rods are bought from a manufacturer who uses a machine to cut rods of length \(L \mathrm {~cm}\), where \(L \sim \mathrm {~N} \left( \mu , 0.2 ^ { 2 } \right)\)

The value of \(\mu\) is such that there is only a \(5 \%\) chance that a rod, selected at random from those supplied to the shop, will have length less than 200 cm .

1. Find the value of \(\mu\) to one decimal place. A customer buys a random sample of 8 of these rods.
2. Find the probability that at least 3 of these rods will have length less than 200 cm . Another customer buys a random sample of 60 of these rods.
3. Using a suitable approximation, find the probability that more than 5 of these rods will have length less than 200 cm .
   

2. The weekly sales, \(S\), in thousands of pounds, of a small business has probability density function $$\mathrm { f } ( s ) = \left\{ \begin{array} { c c } k ( s - 2 ) ( 10 - s ) & 2 < s < 10 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Use algebraic integration to show that \(k = \frac { 3 } { 256 }\)
2. Write down the value of \(\mathrm { E } ( S )\)
3. Use algebraic integration to find the standard deviation of the weekly sales. A week is selected at random.
4. Showing your working, find the probability that this week's sales exceed \(\pounds 7100\) Give your answer to one decimal place. A quarter is defined as 12 consecutive weeks. The discrete random variable \(X\) is the number of weeks in a quarter in which the weekly sales exceed £7100 The manager earns a bonus at the following rates:

   \(\boldsymbol { X }\)	Bonus Earned
   \(X \leqslant 5\)	\(\pounds 0\)
   \(X = 6\)	\(\pounds 1000\)
   \(X \geqslant 7\)	\(\pounds 5000\)

5. Using your answer to part (d), calculate the manager's expected bonus per quarter.

1. At a cafe, customers ordering hot drinks order either tea or coffee.

Of all customers ordering hot drinks, \(80 \%\) order tea and \(20 \%\) order coffee. Of those who order tea, \(35 \%\) take sugar and of those who order coffee \(60 \%\) take sugar.

1. A random sample of 12 customers ordering hot drinks is selected. Find the probability that fewer than 3 of these customers order coffee.
2. 1. A randomly selected customer who orders a hot drink is chosen. Show that the probability that the customer takes sugar is 0.4
   2. Write down the distribution for the number of customers who take sugar from a random sample of \(n\) customers ordering hot drinks.
3. A random sample of 10 customers ordering hot drinks is selected.
   1. Find the probability that exactly 4 of these 10 customers take sugar.
   2. Given that at least 3 of these 10 customers take sugar, find the probability that no more than 6 of these 10 customers take sugar.
4. In a random sample of 150 customers ordering hot drinks, find, using a suitable approximation, the probability that at least half of them take sugar.

4. A bag contains a large number of marbles, each of which is blue or red. A random sample of 3 marbles is taken from the bag. The random variable \(D\) represents the number of blue marbles taken minus the number of red marbles taken. Given that 20\% of the marbles in the bag are blue,

1. show that \(\mathrm { P } ( D = - 1 ) = 0.384\)
2. find the sampling distribution of \(D\)
3. write down the mode of \(D\) Takashi claims that the true proportion of blue marbles is greater than 20\% and tests his claim by selecting a random sample of 12 marbles from the bag.
4. Find the critical region for this test at the 10\% level of significance.
5. State the actual significance level of this test. \includegraphics[max width=\textwidth, alt={}, center]{d2f40cdb-917a-4377-88f4-396766a299e2-15_2255_47_314_37}

1. In the summer Kylie catches a local steam train to work each day. The published arrival time for the train is 10 am.

The random variable \(W\) is the train's actual arrival time minus the published arrival time, in minutes. When the value of \(W\) is positive, the train is late. The cumulative distribution function \(\mathrm { F } ( w )\) is shown in the sketch below. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-06_583_1235_589_349}

1. Specify fully the probability density function \(\mathrm { f } ( w )\) of \(W\).
2. Write down the value of \(\mathrm { E } ( \mathrm { W } )\)
3. Calculate \(\alpha\) such that \(\mathrm { P } ( \alpha \leqslant W \leqslant 1.6 ) = 0.35\) A day is selected at random.
4. Calculate the probability that on this day the train arrives between 1.2 minutes late and 2.4 minutes late. Given that on this day the train was between 1.2 minutes late and 2.4 minutes late,
5. calculate the probability that it was more than 2 minutes late. A random sample of 40 days is taken.
6. Calculate the probability that for at least 10 of these days the train is between 1.2 minutes late and 2.4 minutes late. DO NOT WRITEIN THIS AREA
   

3. A manufacturer produces plates. The proportion of plates that are flawed is \(45 \%\), with flawed plates occurring independently. A random sample of 10 of these plates is selected.

1. Find the probability that the sample contains
   1. fewer than 2 flawed plates,
   2. at least 6 flawed plates.
      (4) George believes that the proportion of flawed plates is not \(45 \%\). To assess his belief George takes a random sample of 120 plates. The random variable \(F\) represents the number of flawed plates found in the sample.
2. Using a normal approximation, find the maximum number of plates, \(c\), and the minimum number of plates, \(d\), such that $$\mathrm { P } ( F \leqslant c ) \leqslant 0.05 \text { and } \mathrm { P } ( F \geqslant d ) \leqslant 0.05$$ where \(F \sim \mathrm {~B} ( 120,0.45 )\) The manufacturer claims that, after a change to the production process, the proportion of flawed plates has decreased. A random sample of 30 plates, taken after the change to the production process, contains 8 flawed plates.
3. Use a suitable hypothesis test, at the \(5 \%\) level of significance, to assess the manufacturer's claim. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{3a781851-e2cc-4379-8b8c-abb3060a6019-11_2255_50_314_34}

1. A research project into food purchases found that \(35 \%\) of people who buy eggs do not buy free range eggs.

A random sample of 30 people who bought eggs is taken. The random variable \(F\) denotes the number of people who do not buy free range eggs.

1. Find \(\mathrm { P } ( F \geqslant 12 )\)
2. Find \(\mathrm { P } ( 8 \leqslant F < 15 )\) A farm shop gives 3 loyalty points with every purchase of free range eggs. With every purchase of eggs that are not free range the farm shop gives 1 loyalty point. A random sample of 30 customers who buy eggs from the farm shop is taken.
3. Find the probability that the total number of points given to these customers is less than 70 The manager of the farm shop believes that the proportion of customers who buy eggs but do not buy free range eggs is more than \(35 \%\) In a survey of 200 customers who buy eggs, 86 do not buy free range eggs. Using a suitable test and a normal approximation,
4. determine, at the \(5 \%\) level of significance, whether there is evidence to support the manager's belief. State your hypotheses clearly.

2. (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 8 < X < 14 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 11\)

1. write down \(\mathrm { P } ( X > 14 )\)
2. find \(\mathrm { P } ( 6 X > a + b )\) (ii) Susie makes a strip of pasta 45 cm long. She then cuts the strip of pasta, at a randomly chosen point, into two pieces. The random variable \(S\) is the length of the shortest piece of pasta.
   1. Write down the distribution of \(S\)
   2. Calculate the probability that the shortest piece of pasta is less than 12 cm long. Susie makes 20 strips of pasta, all 45 cm long, and separately cuts each strip of pasta, at a randomly chosen point, into two pieces.
   3. Calculate the probability that exactly 6 of the pieces of pasta are less than 12 cm long.

Bhavna produces rolls of cloth. She knows that faults occur randomly in her cloth at a mean rate of 1.5 every 15 metres.

1. Find the probability that in 15 metres of her cloth there are
   1. less than 3 faults,
   2. at least 6 faults. Each roll contains 100 metres of Bhavna's cloth.
      She selects 15 rolls at random.
2. Find the probability that exactly 10 of these rolls each have fewer than 13 faults. Bhavna decides to sell her cloth in pieces.
   Each piece of her cloth is 4 metres long.
   The cost to make each piece is \(\pounds 5.00\) She sells each piece of her cloth that contains no faults for \(\pounds 7.40\) She sells each piece of her cloth that contains faults for \(\pounds 2.00\)
3. Find the expected profit that Bhavna will make on each piece of her cloth that she sells.

1. A company produces packets of sunflower seeds. Each packet contains 40 seeds. The company claims that, on average, only 35\% of its sunflower seeds do not germinate.

A packet is selected at random.

1. Using a \(5 \%\) level of significance, find an appropriate critical region for a two-tailed test that the proportion of sunflower seeds that do not germinate is 0.35 You should state your hypotheses clearly and state the probability, which should be as close as possible to \(2.5 \%\), for each tail of your critical region.
2. Write down the actual significance level of this test. Past records suggest that \(2.8 \%\) of the company's sunflower seeds grow to a height of more than 3 metres.
   A random sample of 250 of the company's sunflower seeds is taken and 11 of them grow to a height of more than 3 metres.
3. Using a suitable approximation test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of sunflower seeds that grow to a height of more than 3 metres is now greater than \(2.8 \%\) State your hypotheses clearly.

The probability that a person completes a particular task in less than 15 minutes is 0.4 Jeffrey selects 20 people at random and asks them to complete the task. The random variable, \(X\), represents the number of people who complete the task in less than 15 minutes.

1. Find \(\mathrm { P } ( 5 \leqslant X < 8 )\) Mia takes a random sample of 140 people.
   Using a normal approximation, the probability that fewer than \(n\) of these 140 people complete the task in less than 15 minutes is 0.0239 to 4 decimal places.
2. Find the value of \(n\) Show your working clearly.

1. Sam is a telephone sales representative.

For each call to a customer

• Sam either makes a sale or does not make a sale
• sales are made independently

Past records show that, for each call to a customer, the probability that Sam makes a sale is 0.2

1. Find the probability that Sam makes
   1. exactly 2 sales in 14 calls,
   2. more than 3 sales in 25 calls. Sam makes \(n\) calls each day.
2. Find the minimum value of \(n\)
   1. so that the expected number of sales each day is at least 6
   2. so that the probability of at least 1 sale in a randomly selected day exceeds 0.95

6. A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that \(40 \%\) of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

1. Define the population associated with the magazine.
2. Suggest a suitable sampling frame for the survey.
3. Identify the sampling units.
4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. As a pilot study the editor took a random sample of 25 subscribers.
5. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. In fact only 6 subscribers agreed to the name being changed.
6. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the percentage agreeing to the change is less that the editor believes. The full survey is to be carried out using 200 randomly chosen subscribers.
7. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. \section*{END}

1. The random variables \(R , S\) and \(T\) are distributed as follows

$$R \sim \mathrm {~B} ( 15,0.3 ) , \quad S \sim \mathrm { Po } ( 7.5 ) , \quad T \sim \mathrm {~N} \left( 8,2 ^ { 2 } \right) .$$ Find

1. \(\mathrm { P } ( R = 5 )\),
2. \(\mathrm { P } ( S = 5 )\),
3. \(\mathrm { P } ( T = 5 )\).
   

5. From company records, a manager knows that the probability that a defective article is produced by a particular production line is 0.032 . A random sample of 10 articles is selected from the production line.

1. Find the probability that exactly 2 of them are defective. On another occasion, a random sample of 100 articles is taken.
2. Using a suitable approximation, find the probability that fewer than 4 of them are defective. At a later date, a random sample of 1000 is taken.
3. Using a suitable approximation, find the probability that more than 42 are defective.
   (6)

1. A fair coin is tossed 4 times.

Find the probability that

1. an equal number of head and tails occur
2. all the outcomes are the same,
3. the first tail occurs on the third throw.
   

7. A teacher thinks that \(20 \%\) of the pupils in a school read the Deano comic regularly. He chooses 20 pupils at random and finds 9 of them read the Deano.

1. 1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20\%. State your hypotheses clearly.
   2. State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the \(5 \%\) level.
      (9) The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
2. By combining all 5 samples and using a suitable approximation test, at the \(5 \%\) level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20\%.
3. Comment on your results for the tests in part (a) and part (b).

2. The random variable \(J\) has a Poisson distribution with mean 4.

1. Find \(\mathrm { P } ( J \geqslant 10 )\). The random variable \(K\) has a binomial distribution with parameters \(n = 25 , p = 0.27\).
2. Find \(\mathrm { P } ( K \leqslant 1 )\).

3. For a particular type of plant \(45 \%\) have white flowers and the remainder have coloured flowers. Gardenmania sells plants in batches of 12. A batch is selected at random. Calculate the probability that this batch contains

1. exactly 5 plants with white flowers,
2. more plants with white flowers than coloured ones. Gardenmania takes a random sample of 10 batches of plants.
3. Find the probability that exactly 3 of these batches contain more plants with white flowers than coloured ones. Due to an increasing demand for these plants by large companies, Gardenmania decides to sell them in batches of 50 .
4. Use a suitable approximation to calculate the probability that a batch of 50 plants contains more than 25 plants with white flowers.

2. The probability of a bolt being faulty is 0.3 . Find the probability that in a random sample of 20 bolts there are

1. exactly 2 faulty bolts,
2. more than 3 faulty bolts. These bolts are sold in bags of 20. John buys 10 bags.
3. Find the probability that exactly 6 of these bags contain more than 3 faulty bolts.

6. The probability that a sunflower plant grows over 1.5 metres high is 0.25 . A random sample of 40 sunflower plants is taken and each sunflower plant is measured and its height recorded.

1. Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using
   1. a Poisson approximation,
   2. a Normal approximation.
2. Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer.

A manufacturer supplies DVD players to retailers in batches of 20 . It has \(5 \%\) of the players returned because they are faulty.

1. Write down a suitable model for the distribution of the number of faulty DVD players in a batch. Find the probability that a batch contains
2. no faulty DVD players,
3. more than 4 faulty DVD players.
4. Find the mean and variance of the number of faulty DVD players in a batch.
   

3. The probability of a telesales representative making a sale on a customer call is 0.15 Find the probability that

1. no sales are made in 10 calls,
2. more than 3 sales are made in 20 calls. Representatives are required to achieve a mean of at least 5 sales each day.
3. Find the least number of calls each day a representative should make to achieve this requirement.
4. Calculate the least number of calls that need to be made by a representative for the probability of at least 1 sale to exceed 0.95

A random variable \(X\) has the distribution \(\mathrm { B } ( 12 , p )\).

1. Given that \(p = 0.25\) find
   1. \(\mathrm { P } ( X < 5 )\)
   2. \(\mathrm { P } ( X \geqslant 7 )\)
2. Given that \(\mathrm { P } ( X = 0 ) = 0.05\), find the value of \(p\) to 3 decimal places.
3. Given that the variance of \(X\) is 1.92 , find the possible values of \(p\).