5.01a Permutations and combinations: evaluate probabilities

3. The manager of a gym claimed that the mean age of its customers is 30 years. A random sample of 75 customers is taken and their ages have a mean of 28.2 years and a standard deviation, \(s\), of 8.5 years.

1. Stating your hypotheses clearly and using a 10\% level of significance, test whether or not the manager's claim is supported by the data.
2. Explain the relevance of the Central Limit Theorem to your calculation in part (a).
3. State an additional assumption needed to carry out the test in part (a).

6. The continuous random variable \(Y\) is uniformly distributed over the interval $$[ a - 3 , a + 6 ]$$ where \(a\) is a constant. A random sample of 60 observations of \(Y\) is taken.
Given that \(\bar { Y } = \frac { \sum _ { i = 1 } ^ { 60 } Y _ { i } } { 60 }\)

1. use the Central Limit Theorem to find an approximate distribution for \(\bar { Y }\) Given that the 60 observations of \(Y\) have a sample mean of 13.4
2. find a \(98 \%\) confidence interval for the maximum value that \(Y\) can take.

A college runs academic and vocational courses. The college has 1680 academic students and 2520 vocational students.

1. Describe how a stratified sample of 70 students at the college could be taken. All students at the college take a basic skills test. A random sample of 50 academic students has a mean score of 57 and a variance of 60. An independent random sample of 80 vocational students has a mean score of 62 with a variance of 70
2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance, whether or not the mean basic skills score for vocational students is greater than the mean basic skills score for academic students.
3. Explain the importance of the Central Limit Theorem to the test in part (b).
4. State an assumption that is required to carry out the test in part (b). All the academic students at the college take a basic skills course. Another random sample of 50 academic students and another independent random sample of 80 vocational students retake the basic skills test. The hypotheses used in part (b) are then tested again at the same level of significance. The value of the test statistic \(z\) is now 1.54
5. Comment on the mean basic skills scores of academic and vocational students after taking this course.
6. Considering the outcomes of the tests in part (b) and part (e), comment on the effectiveness of the basic skills course.

Navtej travels to work by train. A train leaves the station every 7 minutes and Navtej's arrival at the station is independent of when the train is due to leave.

1. Write down a suitable model for the distribution of the time, \(T\) minutes, that he has to wait for a train to leave.
2. Find the mean and standard deviation of \(T\) During a 10-week period, Navtej travels to work by train on 46 occasions.
3. Estimate the probability that the mean length of time that he has to wait for a train to leave is between 3.4 and 3.6 minutes.
4. State a necessary assumption for the calculation in part (c).

3.

1. Explain what you understand by the Central Limit Theorem. A garage services hire cars on behalf of a hire company. The garage knows that the lifetime of the brake pads has a standard deviation of 5000 miles. The garage records the lifetimes, \(x\) miles, of the brake pads it has replaced. The garage takes a random sample of 100 brake pads and finds that \(\sum x = 1740000\)
2. Find a 95\% confidence interval for the mean lifetime of a brake pad.
3. Explain the relevance of the Central Limit Theorem in part (b). Brake pads are made to be changed every 20000 miles on average.
   The hire car company complain that the garage is changing the brake pads too soon.
4. Comment on the hire company's complaint. Give a reason for your answer.
   

2. A random sample of size \(n\) is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than \(5 \%\).

6. The continuous random variable \(X\) is uniformly distributed over the interval $$[ a - 1 , a + 5 ]$$ where \(a\) is a constant.
Fifty observations of \(X\) are taken, giving a sample mean of 17.2

1. Use the Central Limit Theorem to find an approximate distribution for \(\bar { X }\).
2. Hence find a 95\% confidence interval for \(a\).

4 The time, \(x\) seconds, spent by each of a random sample of 100 customers at an automatic teller machine (ATM) is recorded. The times are summarised in the table.

1. Calculate estimates for the mean and standard deviation of the time spent at the ATM by a customer.
2. The mean time spent at the ATM by a random sample of \(\mathbf { 3 6 }\) customers is denoted by \(\bar { Y }\).
   1. State why the distribution of \(\bar { Y }\) is approximately normal.
   2. Write down estimated values for the mean and standard error of \(\bar { Y }\).
   3. Hence estimate the probability that \(\bar { Y }\) is less than \(1 \frac { 1 } { 2 }\) minutes.

1 A large bookcase contains two types of book: hardback and paperback. The number of books of each type in each of four subject categories is shown in the table.

1. A book is selected at random from the bookcase. Calculate the probability that the book is:
   1. a paperback;
   2. not science fiction;
   3. science fiction or a hardback;
   4. a thriller, given that it is a paperback.
2. Three books are selected at random, without replacement, from the bookcase. Calculate, to three decimal places, the probability that one is crime, one is romance and one is science fiction.

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.

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}

6. The individual letters of the word STATISTICAL are written on 11 cards which are then shuffled. One card is selected at random. Find the probability that it is

1. a vowel,
2. a T, given that it is a consonant. The 11 cards are then shuffled again and the top three are turned over. Find the probability that
3. all three of the cards have a T on them,
4. at least two of the cards show a vowel.

7. A bag contains 4 red and 2 blue balls, all of the same size. A ball is selected at random and removed from the bag. This is repeated until a blue ball is pulled out of the bag. The random variable \(B\) is the number of balls that have been removed from the bag.

1. Show that \(\mathrm { P } ( B = 2 ) = \frac { 4 } { 15 }\).
2. Find the probability distribution of \(B\).
3. Find \(\mathrm { E } ( B )\). The bag and the same 6 balls are used in a game at a funfair. One ball is removed from the bag at a time and a contestant wins 50 pence if one of the first two balls picked out is blue.
4. What are the expected winnings from playing this game once? For \(\pounds 1\), a contestant gets to play the game three times.
5. What is the expected profit or loss from the three games?

5. The letters of the word DISTRIBUTION are written on separate cards. The cards are then shuffled and the top three are turned over. Let the random variable \(V\) be the number of vowels that are turned over.

1. Show that \(\mathrm { P } ( V = 1 ) = \frac { 21 } { 44 }\).
2. Find the probability distribution of \(V\).
3. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).

5. A College employs 75 teachers, of whom 47 are full-time and the rest are part-time. Of the 39 male teachers at the College, 26 are full-time.

1. Represent this information on a Venn diagram.
2. One teacher is selected at random to be interviewed by an inspector. Find the probability that the teacher chosen
   1. works full-time and is female,
   2. works part-time, given that he is male.
3. Three teachers are selected at random to be observed by an inspector during one day. Find correct to 3 significant figures the probability that
   1. all three teachers chosen work full-time,
   2. at least one of the three teachers chosen is female.

1. There are 16 competitors in a table-tennis competition, 5 of which come from Racknor Comprehensive School. Prizes are awarded to the competitors finishing in each of first, second and third place.

Assuming that all the competitors have an equal chance of success, find the probability that the students from Racknor Comprehensive

1. win no prizes,
2. win the \(1 ^ { \text {st } }\) and \(3 ^ { \text {rd } }\) place prizes but not the \(2 ^ { \text {nd } }\) place prize,
3. win exactly one of the prizes.
   

2 In a quiz, competitors have to match 5 landmarks to the 5 British counties which the landmarks are in. The random variable \(X\) represents the number of correct matches that a competitor gets, assuming that the competitor guesses randomly. The probability distribution of \(X\) is given in the following table.

1. Explain why \(\mathrm { P } ( X = 4 )\) must be 0 .
2. Explain how the value \(\frac { 1 } { 120 }\) for \(\mathrm { P } ( X = 5 )\) is calculated.
3. Draw a graph to illustrate the distribution.
4. Find each of the following.

1 A quiz team of 4 students is to be selected from a group of 7 girls and 5 boys. The team is selected at random from the students in the group. The number of girls in the team is denoted by the random variable \(X\).

1. Show that \(\mathrm { P } ( X = 4 ) = \frac { 7 } { 99 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 99 }\)	\(\frac { 14 } { 99 }\)	\(\frac { 42 } { 99 }\)	\(\frac { 35 } { 99 }\)	\(\frac { 7 } { 99 }\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find each of the following.
   It is decided that the quiz team must have at least 1 girl and at least 1 boy, but the team is still otherwise selected at random.
3. Explain whether \(\mathrm { E } ( X )\) would be smaller than, equal to or larger than the value which you found in part (b).

1 In a game at a fair, players choose 4 countries from a list of 10 countries. The names of all 10 countries are then put in a box and the player selects 4 of them at random. The random variable \(X\) represents the number of countries that match those which the player originally chose.

1. Show that the probability that a randomly selected player matches all 4 countries is \(\frac { 1 } { 210 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 14 }\)	\(\frac { 8 } { 21 }\)	\(\frac { 3 } { 7 }\)	\(\frac { 4 } { 35 }\)	\(\frac { 1 } { 210 }\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find each of the following.
   Find the mean and standard deviation of the player's loss per game.
3. In order to try to attract more customers, the rules will be changed as follows. The game will still cost \(\pounds 1\) to play. The player will get 25 pence back for every country which is matched, plus an additional bonus of \(\pounds 100\) if all four countries are matched. Find the player's mean gain or loss per game with these new rules.

A plumbing company receives call-outs during the working day at an average rate of 2.4 per hour.

1. Find the probability that the company receives exactly 7 call-outs in a randomly selected 3 -hour period of a working day. The company has enough staff to respond to 28 call-outs in an 8 -hour working day.
2. Show that the probability that the company receives more than 28 call-outs in a randomly selected 8 -hour working day is 0.022 to 3 decimal places. In a random sample of 100 working days each of 8 hours,
3. 1. find the expected number of days that the company receives more than 28 call-outs,
   2. find the standard deviation of the number of days that the company receives more than 28 call-outs,
   3. use a Poisson approximation to estimate the probability that the company receives more than 28 call-outs on at least 6 of these days.

Rowan and Alex are both check-in assistants for the same airline. The number of passengers, \(R\), checked in by Rowan during a 30-minute period can be modelled by a Poisson distribution with mean 28

1. Calculate \(\mathrm { P } ( R \geqslant 23 )\) The number of passengers, \(A\), checked in by Alex during a 30-minute period can be modelled by a Poisson distribution with mean 16, where \(R\) and \(A\) are independent. A randomly selected 30-minute period is chosen.
2. Calculate the probability that exactly 42 passengers in total are checked in by Rowan and Alex. The company manager is investigating the rate at which passengers are checked in. He randomly selects 150 non-overlapping 60-minute periods and records the total number of passengers checked in by Rowan and Alex, in each of these 60-minute periods.
3. Using a Poisson approximation, find the probability that for at least 25 of these 60-minute periods Rowan and Alex check in a total of fewer than 80 passengers. On a particular day, Alex complains to the manager that the check-in system is working slower than normal. To see if the complaint is valid the manager takes a random 90-minute period and finds that the total number of people Rowan checks in is 67
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the system is working slower than normal. You should state your hypotheses and conclusion clearly and show your working.

1. The discrete random variable \(X\) has the following probability distribution

1. Find in terms of \(q\)
   1. \(\mathrm { E } ( X )\)
   2. \(\mathrm { E } \left( X ^ { 2 } \right)\) Given that \(\operatorname { Var } ( X ) = 3.66\)
2. show that \(q = 0.3\) In a game, the score is given by the discrete random variable \(X\) Given that games are independent,
3. calculate the probability that after the 4th game has been played, the total score is exactly 20 A round consists of 4 games plus 2 bonus games. The bonus games are only played if after the 4th game has been played the total score is exactly 20 A prize of \(\pounds 10\) is awarded if 6 games are played in a round and the total score for the round is at least 27 Bobby plays 3 rounds.
4. Find the probability that Bobby wins at least \(\pounds 10\)

1. A machine produces cloth. Faults occur randomly in the cloth at a rate of 0.4 per square metre.

The machine is used to produce tablecloths, each of area \(A\) square metres. One of these tablecloths is taken at random. The probability that this tablecloth has no faults is 0.0907

1. Find the value of \(A\) The tablecloths are sold in packets of 20
   A randomly selected packet is taken.
2. Find the probability that more than 1 of the tablecloths in this packet has no faults. A hotel places an order for 100 tablecloths each of area \(A\) square metres.
   The random variable \(X\) represents the number of these tablecloths that have no faults.
3. Find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\)
4. Use a Poisson approximation to estimate \(\mathrm { P } ( X = 10 )\) It is claimed that a new machine produces cloth with a rate of faults that is less than 0.4 per square metre. A piece of cloth produced by this new machine is taken at random.
   The piece of cloth has area 30 square metres and is found to have 6 faults.
5. Stating your hypotheses clearly, use a suitable test to assess the claim made for the new machine. Use a \(5 \%\) level of significance.
6. Write down the \(p\)-value for the test used in part (e).

1. Two car hire companies hire cars independently of each other.

Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.

1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
   In a random sample of 600 new cars produced at the factory,
4. 1. find the mean of the number of cars with a defect,
   2. find the variance of the number of cars with a defect.
5. 1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
   2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}

1. Use Fermat's Little Theorem to find the least positive residue of \(6 ^ { 542 }\) modulo 13
2. Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and Graham, are attending a concert and will sit in a particular row of 7 seats. Find the number of ways they can be seated if

1. there are no restrictions where they sit in the row,
2. Alan, Enid, Felix and Graham sit together,
3. Brenda sits at one end of the row and Graham sits at the other end of the row,
4. Charles and Devindra do not sit together.