5.01a Permutations and combinations: evaluate probabilities

2 The probability generating function, \(\mathrm { G } _ { Y } ( t )\), of the random variable \(Y\) is given by $$G _ { Y } ( t ) = 0.04 + 0.2 t + 0.37 t ^ { 2 } + 0.3 t ^ { 3 } + 0.09 t ^ { 4 }$$

1. Find \(\operatorname { Var } ( Y )\).
   The random variable \(Y\) is the sum of two independent observations of the random variable \(X\).
2. Find the probability generating function of \(X\), giving your answer as a polynomial in \(t\).

3 George throws two coins, \(A\) and \(B\), at the same time. Coin \(A\) is biased so that the probability of obtaining a head is \(a\). Coin \(B\) is biased so that the probability of obtaining a head is \(b\), where \(\mathrm { b } < \mathrm { a }\). The probability generating function of \(X\), the number of heads obtained by George, is \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\). The coefficients of \(t\) and \(t ^ { 2 }\) in \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) are \(\frac { 5 } { 12 }\) and \(\frac { 1 } { 12 }\) respectively.

1. Find the value of \(a\).
   The random variable \(Y\) is the sum of two independent observations of \(X\).
2. Find the probability generating function of \(Y\), giving your answer as a polynomial in \(t\).
3. Find \(\operatorname { Var } ( Y )\).

5 Harry has three coins.

• One coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 3 }\).
• The second coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 4 }\).
• The third coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 5 }\).

The random variable \(X\) is the number of heads that Harry obtains when he throws all three coins together.

1. Find the probability generating function of \(X\).
   Isaac has two fair coins. The random variable \(Y\) is the number of heads that Isaac obtains when he throws both of his coins together. The random variable \(Z\) is the total number of heads obtained when Harry throws his three coins and Isaac throws his two coins.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
3. Use the probability generating function of \(Z\) to find \(E ( Z )\).

8 An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.

1. If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions. Four of the questions, including the one on geometric distributions, are worth 7 marks each, and the remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7-mark questions are the first four questions on the paper, but are arranged in random order. The 9-mark questions are the last four questions, but are arranged in random order. Find the probability that
2. the questions on geometric distributions and on binomial distributions are next to one another,
3. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions.

3 The digits 1, 2, 3, 4 and 5 are arranged in random order, to form a five-digit number.

1. How many different five-digit numbers can be formed?
2. Find the probability that the five-digit number is
   1. odd,
   2. less than 23000 .

1

1. The letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E are arranged in a straight line.
   1. How many different arrangements are possible?
   2. In how many of these arrangements are the letters A and B next to each other?
   3. From the letters \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , two different letters are selected at random. Find the probability that these two letters are A and B .

3 Each of the 7 letters in the word DIVIDED is printed on a separate card. The cards are arranged in a row.

1. How many different arrangements of the letters are possible?
2. In how many of these arrangements are all three Ds together? The 7 cards are now shuffled and 2 cards are selected at random, without replacement.
3. Find the probability that at least one of these 2 cards has D printed on it.

3

1. How many different teams of 7 people can be chosen, without regard to order, from a squad of 15 ?
2. The squad consists of 6 forwards and 9 defenders. How many different teams containing 3 forwards and 4 defenders can be chosen?

3 Five friends, Ali, Bev, Carla, Don and Ed, stand in a line for a photograph.

1. How many different possible arrangements are there if Ali, Bev and Carla stand next to each other?
2. How many different possible arrangements are there if none of Ali, Bev and Carla stand next to each other?
3. If all possible arrangements are equally likely, find the probability that two of Ali, Bev and Carla are next to each other, but the third is not next to either of the other two.

5 A sixth-form class consists of 7 girls and 5 boys. Three students from the class are chosen at random. The number of boys chosen is denoted by the random variable \(X\). Show that

1. \(\quad \mathrm { P } ( X = 0 ) = \frac { 7 } { 44 }\),
2. \(\mathrm { P } ( X = 2 ) = \frac { 7 } { 22 }\). The complete probability distribution of \(X\) is shown in the following table.

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)	\(\frac { 7 } { 44 }\)	\(\frac { 21 } { 44 }\)	\(\frac { 7 } { 22 }\)	\(\frac { 1 } { 22 }\)

3. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

5 A rugby union team consists of 15 players made up of 8 forwards and 7 backs. A manager has to select his team from a squad of 12 forwards and 11 backs.

1. In how many ways can the manager select the forwards?
2. In how many ways can the manager select the team?

6 A band has a repertoire of 12 songs suitable for a live performance. From these songs, a selection of 7 has to be made.

1. Calculate the number of different selections that can be made.
2. Once the 7 songs have been selected, they have to be arranged in playing order. In how many ways can this be done?

8 Jane buys 5 jam doughnuts, 4 cream doughnuts and 3 plain doughnuts.
On arrival home, each of her three children eats one of the twelve doughnuts. The different kinds of doughnut are indistinguishable by sight and so selection of doughnuts is random. Calculate the probabilities of the following events.

1. All 3 doughnuts eaten contain jam.
2. All 3 doughnuts are of the same kind.
3. The 3 doughnuts are all of a different kind.
4. The 3 doughnuts contain jam, given that they are all of the same kind. On 5 successive Saturdays, Jane buys the same combination of 12 doughnuts and her three children eat one each. Find the probability that all 3 doughnuts eaten contain jam on
5. exactly 2 Saturdays out of the 5 ,
6. at least 1 Saturday out of the 5 .

2 Codes of three letters are made up using only the letters A, C, T, G. Find how many different codes are possible

1. if all three letters used must be different,
2. if letters may be repeated.

4 An examination paper consists of three sections.

• Section A contains 6 questions of which the candidate must answer 3
• Section B contains 7 questions of which the candidate must answer 4
• Section C contains 8 questions of which the candidate must answer 5
  1. In how many ways can a candidate choose 3 questions from Section A?
  2. In how many ways can a candidate choose 3 questions from Section A, 4 from Section B and 5 from Section C?

A candidate does not read the instructions and selects 12 questions at random.

Find the probability that they happen to be 3 from Section A, 4 from Section B and 5 from Section C.

4 Peter and Esther visit a restaurant for a three-course meal. On the menu there are 4 starters, 5 main courses and 3 sweets. Peter and Esther each order a starter, a main course and a sweet.

1. Calculate the number of ways in which Peter may choose his three-course meal.
2. Suppose that Peter and Esther choose different dishes from each other.
   (A) Show that the number of possible combinations of starters is 6 .
   (B) Calculate the number of possible combinations of 6 dishes for both meals.
3. Suppose instead that Peter and Esther choose their dishes independently.
   (A) Write down the probability that they choose the same main course.
   (B) Find the probability that they choose different dishes from each other for every course.

1 A girl is choosing tracks from an album to play at her birthday party. The album has 8 tracks and she selects 4 of them.

1. In how many ways can she select the 4 tracks?
2. In how many different orders can she arrange the 4 tracks once she has chosen them?

6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

2. Find the expectation and variance of \(X\).

3 At a dog show, three out of eleven dogs are to be selected for a national competition.

1. Find the number of possible selections.
2. Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.

1 In her purse, Katharine has two \(\pounds 5\) notes, two \(\pounds 10\) notes and one \(\pounds 20\) note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(\pounds X\).

1. (A) Show that \(\mathrm { P } ( X = 10 ) = 0.1\).
   (B) Show that \(\mathrm { P } ( X = 30 ) = 0.2\). The table shows the probability distribution of \(X\).

   \(r\)	10	15	20	25	30
   \(\mathrm { P } ( X = r )\)	0.1	0.4	0.1	0.2	0.2

2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

2. Find the expectation and variance of \(X\).

7 Three independent researchers, \(A , B\) and \(C\), carry out significance tests on the power consumption of a manufacturer's domestic heaters. The power consumption, \(X\) watts, is a normally distributed random variable with mean \(\mu\) and standard deviation 60. Each researcher tests the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 4000\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu > 4000\). Researcher \(A\) uses a sample of size 50 and a significance level of \(5 \%\).

1. Find the critical region for this test, giving your answer correct to 4 significant figures. In fact the value of \(\mu\) is 4020 .
2. Calculate the probability that Researcher \(A\) makes a Type II error.
3. Researcher \(B\) uses a sample bigger than 50 and a significance level of \(5 \%\). Explain whether the probability that Researcher \(B\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
4. Researcher \(C\) uses a sample of size 50 and a significance level bigger than \(5 \%\). Explain whether the probability that Researcher \(C\) makes a Type II error is less than, equal to, or greater than your answer to part (ii).
5. State with a reason whether it is necessary to use the Central Limit Theorem at any point in this question.

2 Two brands of car battery, 'Invincible' and 'Excelsior', have lifetimes which are normally distributed. Invincible batteries have a mean lifetime of 5 years with standard deviation 0.7 years. Excelsior batteries have a mean lifetime of 4.5 years with standard deviation 0.5 years. Random samples of 20 Invincible batteries and 25 Excelsior batteries are selected and the sample mean lifetimes are \(\bar { X } _ { I }\) years and \(\bar { X } _ { E }\) years respectively.

1. State the distributions of \(\bar { X } _ { I }\) and \(\bar { X } _ { E }\).
2. Calculate \(\mathrm { P } \left( \bar { X } _ { I } - \bar { X } _ { E } \geqslant 1 \right)\).

1 A manufacturer of fireworks is investigating the lengths of time for which the fireworks burn. For a particular type of firework this length of time, in minutes, is modelled by the random variable \(T\) with probability density function $$\mathrm { f } ( t ) = k t ^ { 3 } ( 2 - t ) \quad \text { for } 0 < t \leqslant 2$$ where \(k\) is a constant.

1. Show that \(k = \frac { 5 } { 8 }\).
2. Find the modal time.
3. Find \(\mathrm { E } ( T )\) and show that \(\operatorname { Var } ( T ) = \frac { 8 } { 63 }\).
4. A large random sample of \(n\) fireworks of this type is tested. Write down in terms of \(n\) the approximate distribution of \(\bar { T }\), the sample mean time.
5. For a random sample of 100 such fireworks the times are summarised as follows. $$\Sigma t = 145.2 \quad \Sigma t ^ { 2 } = 223.41$$ Find a 95\% confidence interval for the mean time for this type of firework and hence comment on the appropriateness of the model.

6 A City Council comprises 16 Labour members, 14 Conservative members and 6 members of Other parties. A sample of two members was chosen at random to represent the Council at an event. The number of Labour members and the number of Conservative members in this sample are denoted by \(L\) and \(C\) respectively. The joint probability distribution of \(L\) and \(C\) is given in the following table. \(C\)

\(L\)
	0	1	2
0	\(\frac { 1 } { 42 }\)	\(\frac { 16 } { 105 }\)	\(\frac { 4 } { 21 }\)
1	\(\frac { 2 } { 15 }\)	\(\frac { 16 } { 45 }\)	0
2	\(\frac { 13 } { 90 }\)	0	0

1. Verify the two non-zero probabilities in the table for which \(C = 1\).
2. Find the expected number of Conservatives in the sample.
3. Find the expected number of Other members in the sample.
4. Explain why \(L\) and \(C\) are not independent, and state what can be deduced about \(\operatorname { Cov } ( L , C )\).