5.01a Permutations and combinations: evaluate probabilities

4

1. Write down the number of ways of choosing 5 objects from 12 distinct objects.
2. Each possible set of 5 different integers selected from the integers \(1,2 , \ldots , 12\) is obtained, and for each set, the sum of the 5 integers is found. The sum \(S\) can take values between 15 and 50 inclusive. Part of the frequency distribution of \(S\) is shown in the following table, together with the cumulative frequencies.

   S	15	16	17	18	19	20	21	22	23
   Frequency	1	1	2	3	5	7	10	13	17
   Cumulative Frequency	1	2	4	7	12	19	29	42	59

   Use these numbers to determine the critical region for a 1-tail Wilcoxon rank-sum test at the \(2 \%\) significance level when \(m = 5\) and \(n = 7\).
3. A student says that, for a Wilcoxon rank-sum test on samples of size \(m\) and \(n\), where \(m\) and \(n\) are large, the mean and variance of the test statistic \(R _ { m }\) are 200 and \(616 \frac { 2 } { 3 }\) respectively. Show that at least one of these values must be incorrect.

6 A bag contains 6 identical blue counters and 5 identical yellow counters.

1. Three counters are selected at random, without replacement. Find the probability that at least two of the counters are blue. All 11 counters are now arranged in a row in a random order.
2. Find the probability that all the yellow counters are next to each other.
3. Find the probability that no yellow counter is next to another yellow counter.
4. Find the probability that the counters are arranged in such a way that both of the following conditions hold.

2 A project is represented by the activity network and cascade chart below. The table showing the number of workers needed for each activity is incomplete. Each activity needs at least 1 worker. \includegraphics[max width=\textwidth, alt={}, center]{7717b4ca-45ab-4111-9f59-5a3abb04b388-2_202_565_1605_201} \includegraphics[max width=\textwidth, alt={}, center]{7717b4ca-45ab-4111-9f59-5a3abb04b388-2_328_560_1548_820}

1. Complete the table in the Printed Answer Booklet to show the immediate predecessors for each activity.
2. Calculate the latest start time for each non-critical activity. The minimum number of workers needed is 5 .
3. What type of problem (existence, construction, enumeration or optimisation) is the allocation of a number of workers to the activities? There are 8 workers available who can do activities A and B .
4. 1. Find the number of ways that the workers for activity A can be chosen.
   2. When the workers have been chosen for activity A , find the total number of ways of choosing the workers for activity B for all the different possible values of x , where \(\mathrm { x } \geqslant 1\).

3 A para relay team of 4 swimmers needs to be chosen from a group of 7 swimmers.

1. How many ways are there to choose 4 swimmers from 7? There are no restrictions on how many men and how many women are in the team. The group of 7 swimmers consists of 5 men and 2 women.
2. How many ways are there to choose a team with more men than women? The physical impairment of each swimmer is given a score.
   The scores for the swimmers are \(\begin{array} { l l l l l l l } 3 & 4 & 4 & 6 & 7 & 8 & 9 \end{array}\) The total score for the team must be 20 or less.
3. How many different valid teams are possible? The order of the swimmers in the team is now taken into consideration.
4. In total, how many different arrangements are there of valid teams?
5. In how many of these valid teams are the scores of the swimmers in increasing order? For example, 3, 4, 4, 8 but not 4, 3, 4, 8 .

4 The first 20 consecutive positive integers include the 8 prime numbers \(2,3,5,7,11,13,17\) and 19. Emma randomly chooses 5 distinct numbers from the first 20 consecutive positive integers. The order in which Emma chooses the numbers does not matter.

1. Calculate the number of possibilities in which Emma's 5 numbers include exactly 2 prime numbers and 3 non-prime numbers.
2. Calculate the number of possibilities in which Emma's 5 numbers include at least 2 prime numbers. The pairs \(\{ 3,13 \}\) and \(\{ 7,17 \}\) each consist of numbers with a difference of exactly 10 .
3. Calculate the number of possibilities in which Emma's 5 numbers include at least one pair of prime numbers in which the difference between them is exactly 10 . A new set of 20 consecutive positive integers, each with at least two digits, is chosen. This set of 20 numbers contains 5 prime numbers.
4. Use the pigeonhole principle to show that there is at least one pair of these prime numbers for which the difference between them is exactly 10 .

4

1. Show that there are 127 ways to partition a set of 8 distinct elements into two non-empty subsets. A group of 8 people ( \(\mathrm { A } , \mathrm { B } , \ldots\) ) have 8 reserved seats ( \(1,2 , \ldots\) ) on a coach. Seat 1 is reserved for person A , seat 2 for person B , and so on. The reserved seats are labelled but the individual people do not know which seat has been reserved for them. The first 4 people, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , choose their seats at random from the 8 reserved seats.
2. Determine how many different arrangements there are for the seats chosen by \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . The group organiser moves \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D to their correct seats (A in seat \(1 , \mathrm {~B}\) in seat \(2 , \mathrm { C }\) in seat 3 and D in seat 4).
   The other 4 people ( \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H ) then choose their seats at random from the remaining 4 reserved seats ( \(5,6,7\) and 8 ).
3. List the 9 derangements of \(\{ \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H } \}\), where none of these four people is in the seat that has been reserved for them. Suppose, instead, that the 8 people had chosen their seats at random from the 8 reserved seats, without the organiser intervening.
4. Determine the total number of ways in which the seats can be chosen so that 4 of the people are in their correct seats and 4 are not in their correct seats.

3 Bob has been given a pile of five letters addressed to five different people. He has also been given a pile of five envelopes addressed to the same five people. Bob puts one letter in each envelope at random.

1. How many different ways are there to pair the letters with the envelopes?
2. Find the number of arrangements with exactly three letters in the correct envelopes.
3. (a) Show that there are two derangements of the three symbols A , B and C .
   (b) Hence find the number of arrangements with exactly two letters in the correct envelopes. Let \(\mathrm { D } _ { n }\) represent the number of derangements of \(n\) symbols.
4. Explain why \(\mathrm { D } _ { n } = ( n - 1 ) \times \left( \mathrm { D } _ { n - 1 } + \mathrm { D } _ { n - 2 } \right)\).
5. Find the number of ways in which all five letters are in the wrong envelopes.

2. A farmer sells boxes of eggs. The eggs are sold in boxes of 6 eggs and boxes of 12 eggs in the ratio \(n : 1\) A random sample of three boxes is taken.
The number of eggs in the first box is denoted by \(X _ { 1 }\) The number of eggs in the second box is denoted by \(X _ { 2 }\) The number of eggs in the third box is denoted by \(X _ { 3 }\) The random variable \(T = X _ { 1 } + X _ { 2 } + X _ { 3 }\) Given that \(\mathrm { P } ( T = 18 ) = 0.729\)

1. show that \(n = 9\)
2. find the sampling distribution of \(T\) The random variable \(R\) is the range of \(X _ { 1 } , X _ { 2 } , X _ { 3 }\)
3. Using your answer to part (b), or otherwise, find the sampling distribution of \(R\)

6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.

1. Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
2. Suggest a suitable sampling frame.
3. Identify the sampling units. The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70
   When a ball is selected at random, the random variable \(X\) represents the number on the ball where $$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$ A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable \(M\) is the mean of the 2 numbers recorded. Given that $$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
4. show that \(\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }\)

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A bag contains a large number of coins. It only contains 20 p and 50 p coins. A random sample of 3 coins is taken from the bag.

1. List all the possible combinations of 3 coins that might be taken. Let \(\bar { X }\) represent the mean value of the 3 coins taken.
   Part of the sampling distribution of \(\bar { X }\) is given below.

   \(\bar { x }\)	20	\(a\)	\(b\)	50
   \(\mathrm { P } ( \bar { X } = \bar { x } )\)	\(\frac { 4913 } { 8000 }\)	\(c\)	\(d\)	\(\frac { 27 } { 8000 }\)

2. Write down the value of \(a\) and the value of \(b\) The probability of taking a 20p coin at random from the bag is \(p\) The probability of taking a 50p coin at random from the bag is \(q\)
3. Find the value of \(p\) and the value of \(q\)
4. Hence, find the value of \(c\) and the value of \(d\) Let \(M\) represent the mode of the 3 coins taken at random from the bag.
5. Find the sampling distribution of \(M\)

1. A bag contains a large number of balls, each with one of the numbers 1,2 or 5 written on it in the ratio \(2 : 3 : 4\) respectively.

A random sample of 3 balls is taken from the bag.
The random variable \(B\) represents the range of the numbers written on the balls in the sample.

1. Find \(\mathrm { P } ( B = 4 )\)
2. Find the sampling distribution of \(B\).

1. A bag contains 10 counters each with exactly one number written on it.

There are 6 counters with the number 7 on them
There are 3 counters with the number 8 on them
There is 1 counter with the number 9 on it
A random sample of 3 counters is taken from the bag (without replacement).
These counters are then put back in the bag.
This process is then repeated until 20 samples have been taken.
The random variable \(Y\) represents the number of these 20 samples that contain the counter with the number 9 on it.

1. 1. Find the mean of \(Y\)
   2. Find the variance of \(Y\) A random sample of 3 counters is chosen from the bag (without replacement).
2. List all possible samples where the median of the numbers on the 3 counters is 7
3. Find the sampling distribution of the median of the numbers on the 3 counters.

1. An ice cream shop sells a large number of 1 scoop, 2 scoop and 3 scoop ice cream cones to its customers in the ratio \(5 : 2 : 1\)

A random sample of 2 customers at the ice cream shop is taken. Each customer orders a 1 scoop or a 2 scoop or a 3 scoop ice cream cone. Let \(S\) represent the total number of ice cream scoops ordered by these 2 customers.

1. Find the sampling distribution of \(S\) A random sample of \(n\) customers at the ice cream shop is taken. Each customer orders a 1 scoop or a 2 scoop or a 3 scoop ice cream cone. The probability that more than \(n\) scoops of ice cream are ordered by these customers is greater than 0.99
2. Find the smallest possible value of \(n\)
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6.

1. Explain what you understand by the sampling distribution of a statistic. A factory produces beads in bags for craft shops. A small bag contains 40 beads, a medium bag contains 80 beads and a large bag contains 150 beads. The factory produces small, medium and large bags in the ratio 5:3:2 respectively. A random sample of 3 bags is taken from the factory.
2. Find the sampling distribution for the range of the number of beads in the 3 bags in the sample. A random sample of \(n\) sets of 3 bags is taken. The random variable \(Y\) represents the number of these \(n\) sets of 3 bags that have a range of 70
3. Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y = 0 ) < 0.2\)

1. A bag contains a large number of counters.

40\% of the counters are numbered 1 \(35 \%\) of the counters are numbered 2 \(25 \%\) of the counters are numbered 3 In a game Alif draws two counters at random from the bag. His score is 4 times the number on the first counter minus 2 times the number on the second counter.

1. Show that Alif gets a score of 8 with probability 0.0875
2. Find the sampling distribution of Alif's score.
3. Calculate Alif's expected score.

1. A bag contains a large number of counters with one of the numbers 5 , 10 or 20 written on each of them in the ratio \(5 : 2 : a\)

A jar contains a large number of counters with one of the numbers 5 or 10 written on each of them in the ratio \(1 : 3\) One counter is selected at random from the bag and then two counters are selected at random from the jar.
The random variable \(R\) represents the range of the numbers on the 3 counters.
Given that \(\mathrm { P } ( R = 15 ) = \frac { 63 } { 256 }\)

1. by forming and solving an equation in \(a\), show that \(a = 9\)
2. find the sampling distribution of \(R\)

1. A manufacturer makes t -shirts in 3 sizes, small, medium and large.

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts.

20\% of the t -shirts made by the manufacturer are small and sell for \(\pounds 10 30 \%\) of the t -shirts made by the manufacturer are medium and sell for \(\pounds 12\) The rest of the t -shirts made by the manufacturer are large and sell for \(\pounds 15\)

1. Find the mean value of the t -shirts made by the manufacturer. A random sample of 3 t -shirts made by the manufacturer is taken.
2. List all the possible combinations of the individual selling prices of these 3 t-shirts.
3. Find the sampling distribution of the median selling price of these 3 t-shirts. A supermarket receives complaints at a mean rate of 6 per week.

1. State one assumption necessary, in order for a Poisson distribution to be used to model the number of complaints received by the supermarket.
2. Find the probability that, in a given week, there are
   1. fewer than 3 complaints received by the supermarket,
   2. at least 6 complaints received by the supermarket. In a randomly selected week, the supermarket received 12 complaints.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean number of complaints is greater than 6 per week.
   State your hypotheses clearly. Following changes made by the supermarket, it received 26 complaints over a 6-week period.
4. Use a suitable approximation to test whether or not there is evidence that, following the changes, the mean number of complaints received is less than 6 per week. You should state your hypotheses clearly and use a 5\% significance level.
   1. The continuous random variable \(Y\) has cumulative distribution function given by
   $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 0 \\ \frac { 1 } { 21 } y ^ { 2 } & 0 \leqslant y \leqslant k \\ \frac { 2 } { 15 } \left( 6 y - \frac { y ^ { 2 } } { 2 } \right) - \frac { 7 } { 5 } & k < y \leqslant 6 \\ 1 & y > 6 \end{array} \right.$$

1. Find \(\mathrm { P } \left( \left. Y < \frac { 1 } { 4 } k \right\rvert \, Y < k \right)\)
2. Find the value of \(k\)
3. Use algebraic calculus to find \(\mathrm { E } ( Y )\)
   1. The discrete random variable \(X\) is given by
   $$X \sim \mathrm {~B} ( n , p )$$ The value of \(n\) and the value of \(p\) are such that \(X\) can be approximated by a normal random variable \(Y\) where $$Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$$ Given that when using a normal approximation $$\mathrm { P } ( X < 86 ) = 0.2266 \text { and } \mathrm { P } ( X > 97 ) = 0.1056$$

1. show that \(\sigma = 6\)
2. Hence find the value of \(n\) and the value of \(p\)

3. Explain what you understand by

1. a statistic,
2. a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
3. Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) denote the number of screws in each of these packets.
4. List all the possible samples.
5. Find the sampling distribution of \(\bar { Y }\)

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Explain what you understand by

1. a population,
2. a statistic. A researcher took a sample of 100 voters from a certain town and asked them who they would vote for in an election. The proportion who said they would vote for Dr Smith was \(35 \%\).
3. State the population and the statistic in this case.
4. Explain what you understand by the sampling distribution of this statistic.
   

Explain what you understand by

1. a population,
2. a statistic. A questionnaire concerning attitudes to classes in a college was completed by a random sample of 50 students. The students gave the college a mean approval rating of 75\%.
3. Identify the population and the statistic in this situation.
4. Explain what you understand by the sampling distribution of this statistic.
   

6. A bag contains a large number of balls. 65\% are numbered 1 35\% are numbered 2 A random sample of 3 balls is taken from the bag.
Find the sampling distribution for the range of the numbers on the 3 selected balls.

1. A bag contains a large number of counters. A third of the counters have a number 5 on them and the remainder have a number 1 .

A random sample of 3 counters is selected.

1. List all possible samples.
2. Find the sampling distribution for the range.
   

A bag contains a large number of \(1 \mathrm { p } , 2 \mathrm { p }\) and 5 p coins. \(50 \%\) are 1 p coins \(20 \%\) are \(2 p\) coins
30\% are 5p coins
A random sample of 3 coins is chosen from the bag.

1. List all the possible samples of size 3 with median 5p.
2. Find the probability that the median value of the sample is 5 p .
3. Find the sampling distribution of the median of samples of size 3
   

6. A bag contains a large number of counters with one of the numbers 4,6 or 8 written on each of them in the ratio \(5 : 3 : 2\) respectively. A random sample of 2 counters is taken from the bag.

1. List all the possible samples of size 2 that can be taken. The random variable \(M\) represents the mean value of the 2 counters.
   Given that \(\mathrm { P } ( M = 4 ) = \frac { 1 } { 4 }\) and \(\mathrm { P } ( M = 8 ) = \frac { 1 } { 25 }\)
2. find the sampling distribution for \(M\). A sample of \(n\) sets of 2 counters is taken. The random variable \(Y\) represents the number of these \(n\) sets that have a mean of 8
3. Calculate the minimum value of \(n\) such that \(\mathrm { P } ( Y \geqslant 1 ) > 0.9\)