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

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.
   

4. The random variable \(X \sim \mathrm {~B} ( 150,0.02 )\). Use a suitable approximation to estimate \(\mathrm { P } ( X > 7 )\).

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.

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 \(p\).
   

4. A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that \(20 \%\) of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is

1. at least 3,
2. fewer than 2 . One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,
3. use a suitable approximation to find the probability that there are enough first class stamps.
4. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid.

2. A fair coin is spun 6 times and the random variable \(T\) represents the number of tails obtained.

1. Give two reasons why a binomial model would be a suitable distribution for modelling \(T\).
2. Find \(\mathrm { P } ( T = 5 )\)
3. Find the probability of obtaining more tails than heads. A second coin is biased such that the probability of obtaining a head is \(\frac { 1 } { 4 }\) This second coin is spun 6 times.
4. Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.

4 A research student is investigating the number of children who are girls in families with 4 children. The table below shows her results for 200 such families. The research student suggests that a binomial distribution with \(p = \frac { 1 } { 2 }\) could be a suitable model for the number of children who are girls in a family of 4 children.

1. Using her results and a \(5 \%\) significance level, test the research student's claim. You should state your hypotheses, expected frequencies, test statistic and the critical value used. The research student decides to refine the model and retains the idea of using a binomial distribution but does not specify the probability that the child is a girl.
2. Use the data in the table to show that the probability that a child is a girl is 0.45 The research student uses the probability from part (b) to calculate a new set of expected frequencies, none of which are less than 5
   The statistic \(\sum \frac { ( O - E ) ^ { 2 } } { E }\) is evaluated and found to be 2.47
3. Test, at the \(5 \%\) significance level, whether using a binomial distribution is suitable to model the number of children who are girls in a family of 4 children. You should state your hypotheses and the critical value used.

6. Bags of \(\pounds 1\) coins are paid into a bank. Each bag contains 20 coins. The bank manager believes that \(5 \%\) of the \(\pounds 1\) coins paid into the bank are fakes. He decides to use the distribution \(X \sim \mathrm {~B} ( 20,0.05 )\) to model the random variable \(X\), the number of fake \(\pounds 1\) coins in each bag.

1. State the assumptions necessary for the binomial distribution to be an appropriate model in this case. The bank manager checks a random sample of 150 bags of \(\pounds 1\) coins and records the number of fake coins found in each bag. His results are summarised in Table 1. \begin{table}[h]

   Number of fake coins in each bag	0	1	2	3	4 or more
   Observed frequency	43	62	26	13	6
   Expected frequency	53.8	56.6	\(r\)	8.9	\(s\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Calculate the values of \(r\) and \(s\), giving your answers to 1 decimal place.
3. Carry out a hypothesis test, at the \(5 \%\) significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly. Question 6 parts (d) and (e) are continued on page 24 The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than \(5 \%\). She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. \begin{table}[h]

   Number of fake coins in each bag	0	1	2	3	4 or more
   Observed frequency	43	62	26	13	6
   Expected frequency	44.5	55.7	33.2	12.5	4.1

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
4. Explain why there are 2 degrees of freedom in this case.
5. Given that she obtains a \(\chi ^ { 2 }\) test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a \(5 \%\) level of significance and state your hypotheses clearly.

5. A research station is doing some work on the germination of a new variety of genetically modified wheat. They planted 120 rows containing 7 seeds in each row.
The number of seeds germinating in each row was recorded. The results are as follows

1. Write down two reasons why a binomial distribution may be a suitable model.
2. Show that the probability of a randomly selected seed from this sample germinating is 0.6 The research station used a binomial distribution with probability 0.6 of a seed germinating. The expected frequencies were calculated to 2 decimal places. The results are as follows

   Number of seeds germinating in each row	0	1	2	3	4	5	6	7
   Expected number of rows	0.20	2.06	\(s\)	23.22	\(t\)	31.35	15.68	3.36

3. Find the value of \(s\) and the value of \(t\).
4. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the data can be modelled by a binomial distribution.

6 Plastic clothes pegs are made in various colours.
The number of red pegs may be modelled by a binomial distribution with parameter \(p\) equal to 0.2 . The contents of packets of 50 pegs of mixed colours may be considered to be random samples.

1. Determine the probability that a packet contains:
   1. less than or equal to 15 red pegs;
   2. exactly 10 red pegs;
   3. more than 5 but fewer than 15 red pegs.
2. Sly, a student, claims to have counted the number of red pegs in each of 100 packets of 50 pegs. From his results the following values are calculated. Mean number of red pegs per packet \(= 10.5\) Variance of number of red pegs per packet \(= 20.41\) Comment on the validity of Sly's claim.

7 A travel agency in Tunisia offers customers a 3-day tour into the Sahara desert by either coach or minibus.

1. The agency accepts bookings from 50 customers for seats on the coach. The probability that a customer, who has booked a seat on the coach, will not turn up to claim the seat is 0.08 , and may be assumed to be independent of the behaviour of other customers. Determine the probability that, of the customers who have booked a seat on the coach:
   1. two or more will not turn up;
   2. three or more will not turn up.
2. The agency accepts bookings from 15 customers for seats on the minibus. The probability that a customer, who has booked a seat on the minibus, will not turn up to claim the seat is 0.025 , and may be assumed to be independent of the behaviour of other customers. Calculate the probability that, of the customers who have booked a seat on the minibus:
   1. all will turn up;
   2. one or more will not turn up.
3. The coach has 48 seats and the minibus has 14 seats. If 14 or fewer customers who have booked seats on the minibus turn up, they will be allocated a seat on the minibus. If all 15 customers who have booked seats on the minibus turn up, one will be allocated a seat on the coach. This will leave only 47 seats available for the 50 customers who have booked seats on the coach. Use your results from parts (a) and (b) to calculate the probability that there will be seats available on the coach for all those who turn up having booked such seats.
   (4 marks)

7 The proportion of passengers who use senior citizen bus passes to travel into a particular town on 'Park \& Ride' buses between 9.30 am and 11.30 am on weekdays is 0.45 . It is proposed that, when there are \(n\) passengers on a bus, a suitable model for the number of passengers using senior citizen bus passes is the distribution \(\mathrm { B } ( n , 0.45 )\).

1. Assuming that this model applies to the 10.30 am weekday 'Park \& Ride' bus into the town:
   1. calculate the probability that, when there are \(\mathbf { 1 6 }\) passengers, exactly 3 of them are using senior citizen bus passes;
   2. determine the probability that, when there are \(\mathbf { 2 5 }\) passengers, fewer than 10 of them are using senior citizen bus passes;
   3. determine the probability that, when there are \(\mathbf { 4 0 }\) passengers, at least 15 but at most 20 of them are using senior citizen bus passes;
   4. calculate the mean and the variance for the number of passengers using senior citizen bus passes when there are \(\mathbf { 5 0 }\) passengers.
2. 1. Give a reason why the proposed model may not be suitable.
   2. Give a different reason why the proposed model would not be suitable for the number of passengers using senior citizen bus passes to travel into the town on the 7.15 am weekday 'Park \& Ride' bus.

4 Clay pigeon shooting is the sport of shooting at special flying clay targets with a shotgun.

1. Rhys, a novice, uses a single-barrelled shotgun. The probability that he hits a target is 0.45 , and may be assumed to be independent from target to target. Determine the probability that, in a series of shots at 15 targets, he hits:
   1. at most 5 targets;
   2. more than 10 targets;
   3. exactly 6 targets;
   4. at least 5 but at most 10 targets.
2. Sasha, an expert, uses a double-barrelled shotgun. She shoots at each target with the gun's first barrel and, only if she misses, does she then shoot at the target with the gun's second barrel. The probability that she hits a target with a shot using her gun's first barrel is 0.85 . The conditional probability that she hits a target with a shot using her gun's second barrel, given that she has missed the target with a shot using her gun's first barrel, is 0.80 . Assume that Sasha's shooting is independent from target to target.
   1. Show that the probability that Sasha hits a target is 0.97 .
   2. Determine the probability that, in a series of shots at 50 targets, Sasha hits at least 48 targets.
   3. In a series of shots at 80 targets, calculate the mean number of times that Sasha shoots at targets with her gun's second barrel.

4 The records at a passport office show that, on average, 15 per cent of photographs that accompany applications for passport renewals are unusable. Assume that exactly one photograph accompanies each application.

1. Determine the probability that in a random sample of 40 applications:
   1. exactly 6 photographs are unusable;
   2. at most 5 photographs are unusable;
   3. more than 5 but fewer than 10 photographs are unusable.
2. Calculate the mean and the standard deviation for the number of photographs that are unusable in a random sample of \(\mathbf { 3 2 }\) applications.
3. Mr Stickler processes 32 applications each day. His records for the previous 10 days show that the numbers of photographs that he deemed unusable were $$\begin{array} { l l l l l l l l l l } 8 & 6 & 10 & 7 & 9 & 7 & 8 & 9 & 6 & 7 \end{array}$$ By calculating the mean and the standard deviation of these values, comment, with reasons, on the suitability of the \(\mathrm { B } ( 32,0.15 )\) model for the number of photographs deemed unusable each day by Mr Stickler.

3 Stopoff owns a chain of hotels. Guests are presented with the bills for their stays when they check out.

1. Assume that the number of bills that contain errors may be modelled by a binomial distribution with parameters \(n\) and \(p\), where \(p = 0.30\). Determine the probability that, in a random sample of 40 bills:
   1. at most 10 bills contain errors;
   2. at least 15 bills contain errors;
   3. exactly 12 bills contain errors.
2. Calculate the mean and the variance for each of the distributions \(\mathrm { B } ( 16,0.20 )\) and \(B ( 16,0.125 )\).
3. Stan, who is a travelling salesperson, always uses Stopoff hotels. He holds one of its diamond customer cards and so should qualify for special customer care. However, he regularly finds errors in his bills when he checks out. Each month, during a 12-month period, Stan stayed in Stopoff hotels on exactly 16 occasions. He recorded, each month, the number of occasions on which his bill contained errors. His recorded values were as follows. $$\begin{array} { l l l l l l l l l l l l } 2 & 1 & 4 & 3 & 1 & 3 & 0 & 3 & 1 & 0 & 5 & 1 \end{array}$$
   1. Calculate the mean and the variance of these 12 values.
   2. Hence state with reasons which, if either, of the distributions \(\mathrm { B } ( 16,0.20 )\) and \(B ( 16,0.125 )\) is likely to provide a satisfactory model for these 12 values.

6 Each weekday, Monday to Friday, Trina catches a train from her local station. She claims that the probability that the train arrives on time at the station is 0.4 , and that the train's arrival time is independent from day to day.

1. Assuming her claims to be true, determine the probability that the train arrives on time at the station:
   1. on at most 3 days during a 2 -week period ( 10 days);
   2. on more than 10 days but fewer than 20 days during an 8-week period.
2. 1. Assuming Trina's claims to be true, determine the mean and standard deviation for the number of times during a week (5 days) that the train arrives on time at the station.
   2. Each week, for a period of 13 weeks, Trina's travelling colleague, Suzie, records the number of times that the train arrives on time at the station. Suzie's results are

      2

      2

      4

      1

      2

      3

      3

      2

      2

      0

      3

      2

      0

      Calculate the mean and standard deviation of these values.
   3. Hence comment on the likely validity of Trina's claims.

6 For the adult population of the UK, 35 per cent of men and 29 per cent of women do not wear glasses or contact lenses.

1. Determine the probability that, in a random sample of 40 men:
   1. at most 15 do not wear glasses or contact lenses;
   2. more than 10 but fewer than 20 do not wear glasses or contact lenses.
2. Calculate the probability that, in a random sample of 10 women, exactly 3 do not wear glasses or contact lenses.
3. 1. Calculate the mean and the variance for the number who do wear glasses or contact lenses in a random sample of 20 women.
   2. The numbers wearing glasses or contact lenses in 10 groups, each of 20 women, had a mean of 16.5 and a variance of 2.50. Comment on the claim that these 10 groups were not random samples.

7 Mr Alott and Miss Fewer work in a postal sorting office.

1. The number of letters per batch, \(R\), sorted incorrectly by Mr Alott when sorting batches of 50 letters may be modelled by the distribution \(\mathrm { B } ( 50,0.15 )\). Determine:
   1. \(\mathrm { P } ( R < 10 )\);
   2. \(\mathrm { P } ( 5 \leqslant R \leqslant 10 )\).
2. It is assumed that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
   1. Calculate the probability that, when sorting a batch of \(\mathbf { 2 2 }\) letters, Miss Fewer sorts exactly 2 letters incorrectly.
   2. Calculate the probability that, when sorting a batch of \(\mathbf { 3 5 }\) letters, Miss Fewer sorts at least 1 letter incorrectly.
   3. Calculate the mean and the variance for the number of letters sorted correctly by Miss Fewer when she sorts a batch of \(\mathbf { 1 2 0 }\) letters.
   4. Miss Fewer sorts a random sample of 20 batches, each containing 120 letters. The number of letters sorted correctly per batch has a mean of 112.8 and a variance of 56.86 . Comment on the assumptions that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
      \includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-15_2484_1709_223_153}

4 In a certain country, 15 per cent of the male population is left-handed.

1. Determine the probability that, in a random sample of 50 males from this country:
   1. at most 10 are left-handed;
   2. at least 5 are left-handed;
   3. more than 6 but fewer than 12 are left-handed.
2. In the same country, 11 per cent of the female population is left-handed. Calculate the probability that, in a random sample of 35 females from this country, exactly 4 are left-handed.
3. A sample of 2000 people is selected at random from the population of the country. The proportion of males in the sample is 52 per cent. How many people in the sample would you expect to be left-handed?
   \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-09_2484_1709_223_153}