2.04c Calculate binomial probabilities

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.

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 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}

6 An amateur tennis club purchases tennis balls that have been used previously in professional tournaments. The probability that each such ball fails a standard bounce test is 0.15 . The club purchases boxes each containing 10 of these tennis balls. Assume that the 10 balls in any box represent a random sample.

1. Determine the probability that the number of balls in a box which fail the bounce test is:
   1. at most 2 ;
   2. at least 2;
   3. more than 1 but fewer than 5 .
2. Determine the probability that, in \(\mathbf { 5 }\) boxes, the total number of balls which fail the bounce test is:
   1. more than 5 ;
   2. at least 5 but at most 10 .

4 A survey of the 640 properties on an estate was undertaken. Part of the information collected related to the number of bedrooms and the number of toilets in each property. This information is shown in the table.

1. A property on the estate is selected at random. Find, giving your answer to three decimal places, the probability that the property has:
   1. exactly 3 bedrooms;
   2. at least 2 toilets;
   3. exactly 3 bedrooms and at least 2 toilets;
   4. at most 3 bedrooms, given that it has exactly 2 toilets.
2. Use relevant answers from part (a) to show that the number of toilets is not independent of the number of bedrooms.
3. Three properties are selected at random from those on the estate which have exactly 3 bedrooms. Calculate the probability that one property has 2 toilets, one has 3 toilets and the other has at least 4 toilets. Give your answer to three decimal places.

6 A bin contains a very large number of paper clips of different colours. The proportion of each colour is shown in the table.

1. Packets are filled from the bin. Each filled packet contains exactly 30 paper clips which may be considered to be a random sample. Use binomial distributions to determine the probability that a filled packet contains:
   1. exactly 2 purple paper clips;
   2. a total of more than 10 red or purple paper clips;
   3. at least 5 but at most 10 green paper clips.
2. Jumbo packets are also filled from the bin. Each filled jumbo packet contains exactly 100 paper clips.
   1. Assuming that the number of paper clips in a jumbo packet may be considered to be a random sample, calculate the mean and the variance of the number of red paper clips in a filled jumbo packet.
   2. It is claimed that the proportion of red paper clips in the bin is greater than 0.22 and that jumbo packets do not contain random samples of paper clips. An analysis of the number of red paper clips in each of a random sample of 50 filled jumbo packets resulted in a mean of 22.1 and a standard deviation of 4.17. Comment, with numerical justification, on each of the two claims.

6 The probability that an online order from a supermarket chain has at least one item missing when delivered is 0.06 . Online orders are 'incomplete' if they contain substitute items and/or have at least one item missing when delivered. The probability that an order is incomplete is 0.15 .

1. Calculate the probability that exactly 2 out of a random sample of 26 online orders have at least one item missing when delivered.
2. Determine the probability that the number of incomplete orders in a random sample of 50 online orders is:
   1. fewer than 10 ;
   2. more than 5;
   3. more than 6 but fewer than 12 .
3. Farokh, the manager of one of the supermarket's stores, examines 50 randomly selected online orders from each of a random sample of 100 of the store's customers. He records, for each of the 50 orders, the number, \(x\), that were incomplete. His summarised results, correct to three significant figures, for the 100 customers selected are $$\bar { x } = 4.33 \text { and } s ^ { 2 } = 3.94$$ Use this information to compare the performance of the store managed by Farokh with that of the supermarket chain as a whole.
   [0pt] [5 marks]

5 An analysis of the number of vehicles registered by each household within a city resulted in the following information.

1. A random sample of 30 households within the city is selected. Use a binomial distribution with \(n = 30\), together with relevant information from the table in each case, to find the probability that the sample contains:
   1. exactly 3 households with no registered vehicles;
   2. at most 5 households with three or more registered vehicles;
   3. more than 10 households with at least two registered vehicles;
   4. more than 5 households but fewer than 10 households with exactly two registered vehicles.
2. If a random sample of \(\mathbf { 1 5 0 }\) households within the city were to be selected, estimate the mean and the variance for the number of households in the sample that would have either one or two registered vehicles.
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-16_1075_1707_1628_153}

3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.

1. Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
   1. aged between 41 and 65;
   2. aged 66 or over and charged at band 2;
   3. aged between 19 and 40 and charged at most at band 1;
   4. aged 41 or over, given that the patient was charged at band 2;
   5. charged at least at band 2, given that the patient was not aged 66 or over.
2. Four patients at this dental surgery, not included in the above 500 patients, are selected at random. Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.
   [0pt] [5 marks]

6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.

1. A sample of 50 loom bands is selected at random from the box. Use a binomial distribution with \(n = 50\), together with relevant information from the table, to estimate the probability that this sample contains:
   1. exactly 4 red loom bands;
   2. at most 10 yellow loom bands;
   3. at least 30 blue or green loom bands;
   4. more than 35 but fewer than 45 loom bands that are neither yellow nor white.
2. The random variable \(R\) denotes the number of red loom bands in a random sample of \(\mathbf { 3 0 0 }\) loom bands selected from the box. Estimate values for the mean and the variance of \(R\).
   [0pt] [2 marks]

3. The time it takes girls aged 15 to complete an obstacle course is found to be normally distributed with a mean of 21.5 minutes and a standard deviation of 2.2 minutes.

1. Find the probability that a randomly chosen 15 year-old girl completes the course in less than 25 minutes. A 13 year-old girl completes the course in exactly 19 minutes.
2. What percentage of 15 year-old girls would she beat over the course? Anyone completing the course in less than 20 minutes is presented with a certificate of achievement. Three friends all complete the course one afternoon.
3. What is the probability that exactly two of them get certificates?

1 The number of cars passing a speed camera on a main road between 9.30 am and 11.30 am may be modelled by a Poisson distribution with a mean rate of 2.6 per minute.

1. 1. Write down the distribution of \(X\), the number of cars passing the speed camera during a 5-minute interval between 9.30 am and 11.30 am .
   2. Determine \(\mathrm { P } ( X = 20 )\).
   3. Determine \(\mathrm { P } ( 6 \leqslant X \leqslant 18 )\).
2. Give two reasons why a Poisson distribution with mean 2.6 may not be a suitable model for the number of cars passing the speed camera during a 1 -minute interval between 8.00 am and 9.30 am on weekdays.
3. When \(n\) cars pass the speed camera, the number of cars, \(Y\), that exceed 60 mph may be modelled by the distribution \(\mathrm { B } ( n , 0.2 )\). Given that \(n = 20\), determine \(\mathrm { P } ( Y \geqslant 5 )\).
4. Stating a necessary assumption, calculate the probability that, during a given 5-minute interval between 9.30 am and 11.30 am , exactly 20 cars pass the speed camera of which at least 5 are exceeding 60 mph .

Lupin seeds are sold in packets of 15 . On average, 9 seeds in a packet are green and 6 are red. Find, to 2 decimal places, the probability that in any particular packet there are

1. less than 2 red seeds,
2. more red than green seeds. The seeds from 10 packets are then combined together.
3. Use a suitable approximation to find the probability that the total number of green seeds is more than 100 .
   

4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).

1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
2. Evaluate this probability when \(p = 0.6\).

5. In a certain school, \(32 \%\) of Year 9 pupils are left-handed. A random sample of 10 Year 9 pupils is chosen.

1. Find the probability that none are left-handed.
2. Find the probability that at least two are left-handed.
3. Use a suitable approximation to find the probability of getting more than 5 but less than 15 left-handed pupils in a group of 35 randomly selected Year 9 pupils.
   Explain what adjustment is necessary when using this approximation. \section*{STATISTICS 2 (A) TEST PAPER 3 Page 2}

3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.

1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
3. Determine the expected number of answers that Sarah will get right.
4. Find the probability that Sarah gets more than 25 correct answers out of 30.