5.01a Permutations and combinations: evaluate probabilities

6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-05_858_980_573_585}

1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).

   \(r\)	0	1	2	3
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
3. Find the value of \(k\).
4. Calculate the mean and variance of \(R\).

7 A committee of 7 people is to be chosen at random from 18 volunteers.

1. In how many different ways can the committee be chosen? The 18 volunteers consist of 5 people from Gloucester, 6 from Hereford and 7 from Worcester. The committee is to be chosen randomly. Find the probability that the committee will
2. consist of 2 people from Gloucester, 2 people from Hereford and 3 people from Worcester,
3. include exactly 5 people from Worcester,
4. include at least 2 people from each of the three cities. 1 Jenny and John are each allowed two attempts to pass an examination.
5. Jenny estimates that her chances of success are as follows.
   • The probability that she will pass on her first attempt is \(\frac { 2 } { 3 }\).
   • If she fails on her first attempt, the probability that she will pass on her second attempt is \(\frac { 3 } { 4 }\). Calculate the probability that Jenny will pass.
   • John estimates that his chances of success are as follows.
   • The probability that he will pass on his first attempt is \(\frac { 2 } { 3 }\).
   • Overall, the probability that he will pass is \(\frac { 5 } { 6 }\).
   Calculate the probability that if John fails on his first attempt, he will pass on his second attempt. 2 For each of 50 plants, the height, \(h \mathrm {~cm}\), was measured and the value of ( \(h - 100\) ) was recorded. The mean and standard deviation of \(( h - 100 )\) were found to be 24.5 and 4.8 respectively.
6. Write down the mean and standard deviation of \(h\). The mean and standard deviation of the heights of another 100 plants were found to be 123.0 cm and 5.1 cm respectively.
7. Describe briefly how the heights of the second group of plants compare with the first.
8. Calculate the mean height of all 150 plants. 3 In Mr Kendall's cupboard there are 3 tins of baked beans and 2 tins of pineapple. Unfortunately his daughter has removed all the labels for a school project and so the tins are identical in appearance. Mr Kendall wishes to use both tins of pineapple for a fruit salad. He opens tins at random until he has opened the two tins of pineapples. Let \(X\) be the number of tins that Mr Kendall opens.
9. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 5 }\).
10. The probability distribution of \(X\) is given in the table below.

    \(x\)	2	3	4	5
    \(\mathrm { P } ( X = x )\)	\(\frac { 1 } { 10 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 2 } { 5 }\)

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

2 John works from home. The number of business letters, \(X\), that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, \(Y\), that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.

1. Find, for a given weekday:
   1. \(\mathrm { P } ( X < 4 )\);
   2. \(\quad \mathrm { P } ( Y = 4 )\).
2. 1. Assuming that \(X\) and \(Y\) are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
   2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
3. The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
   1. Calculate the mean and the variance of these data.
   2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.

5 Joanne has 10 identically-shaped discs, of which 1 is blue, 2 are green, 3 are yellow and 4 are red. She places the 10 discs in a bag and asks her friend David to play a game by selecting, at random and without replacement, two discs from the bag.

1. Show that:
   1. the probability that the two discs selected are the same colour is \(\frac { 2 } { 9 }\);
   2. the probability that exactly one of the two discs selected is blue is \(\frac { 1 } { 5 }\).
2. Using the discs, Joanne plays the game with David, under the following conditions: If the two discs selected by David are the same colour, she will pay him 135p. If exactly one of the two discs selected by David is blue, she will pay him 145p. Otherwise David will pay Joanne 45p.
   1. When a game is played, \(X\) is the amount, in pence, won by David. Construct the probability distribution for \(X\), in the form of a table.
   2. Show that \(\mathrm { E } ( X ) = 33\).
3. Joanne modifies the game so that the amount per game, \(Y\) pence, that she wins may be modelled by $$Y = 104 - 3 X$$
   1. Determine how much Joanne would expect to win if the game is played 100 times.
   2. Calculate the standard deviation of \(Y\), giving your answer to the nearest 1 p .

1 In a promotion for a new type of cereal, a toy dinosaur is included in each pack. There are three different types of dinosaur to collect. They are distributed, with equal probability, randomly and independently in the packs. Sam is trying to collect all three of the dinosaurs.

1. Find the probability that Sam has to open only 3 packs in order to collect all three dinosaurs. Sam continues to open packs until she has collected all three dinosaurs, but once she has opened 6 packs she gives up even if she has not found all three. The random variable \(X\) represents the number of packs which Sam opens.
2. Complete the table below, using the copy in the Printed Answer Booklet, to show the probability distribution of \(X\).

   \(r\)	3	4	5	6
   \(\mathrm { P } ( X = r )\)		\(\frac { 2 } { 9 }\)	\(\frac { 14 } { 81 }\)

   \section*{(iii) In this question you must show detailed reasoning.} Find
   • \(\mathrm { E } ( X )\) and
   • \(\operatorname { Var } ( X )\).

2 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other.
2. Find the probability that no two men are next to one another.

12 Two players, \(A\) and \(B\), are taking turns to shoot at a basket with a basketball. The winner of this game is the first player to score a basket. The probability that \(A\) scores a basket with any shot is \(\frac { 1 } { 4 }\) and the probability that \(B\) scores a basket with any shot is \(\frac { 1 } { 5 }\). Each shot is independent of all other shots. \(A\) shoots first.

1. Find
   1. the probability that \(B\) wins with his first shot,
   2. the probability that \(A\) wins with his second shot,
   3. the probability that \(A\) wins the game.
   4. \(R\) is the total number of shots taken by \(A\) and \(B\) up to and including the shot that scores a basket.
      (a) Show that the probability generating function of \(R\) is given by $$\mathrm { G } ( t ) = \frac { 5 t + 3 t ^ { 2 } } { 4 \left( 5 - 3 t ^ { 2 } \right) }$$ (b) Hence find \(\mathrm { E } ( R )\).

4 In one department of a firm, four employees are selected for promotion from a staff of eighteen.

1. In how many ways can four employees be selected? It is known that throughout the firm 5\% of those selected for promotion decline it.
2. If 100 employees are randomly selected for promotion in the firm, calculate the number expected to decline promotion.
3. If 20 employees are selected at random for promotion, use the binomial distribution to find the probability that fewer than five employees will decline promotion.

5 A game is played with cards, each of which has a single digit printed on it. Eleanor has 7 cards with the digits \(1,1,2,3,4,5,6\) on them.

1. How many different 7-digit numbers can be made by arranging Eleanor's cards?
2. Eleanor is going to select 5 of the 7 cards and use them to form a 5 -digit number. How many different 5-digit numbers are possible?

6

1. Verify that \(\left( 1 - t ^ { 6 } \right) = ( 1 - t ) \left( 1 + t + t ^ { 2 } + t ^ { 3 } + t ^ { 4 } + t ^ { 5 } \right)\).
2. An unbiased six-faced die is rolled \(r\) times. Show that the probability generating function for the total score is $$\left[ \frac { t \left( 1 - t ^ { 6 } \right) } { 6 ( 1 - t ) } \right] ^ { r }$$
3. Hence show that the probability of the total score being ( \(r + 3\) ) is $$\left( \frac { 1 } { 6 } \right) ^ { r + 1 } r ( r + 1 ) ( r + 2 )$$

2

1. A music club has 200 members. 75 members play the piano, 130 members like Elgar, and 30 members do not play the piano, nor do they like Elgar.
   1. Calculate the probability that a member chosen at random plays the piano but does not like Elgar.
   2. Calculate the probability that a member chosen at random plays the piano given that this member likes Elgar.
2. The music club is organising a concert. The programme is to consist of 7 pieces of music which are to be selected from 9 classical pieces and 6 modern pieces. Find the number of different concert programmes than can be produced if
   1. there are no restrictions,
   2. the programme must consist of 5 classical pieces and 2 modern pieces,
   3. there are to be more modern pieces than classical pieces.

4 The letters of the word 'STATISTICS' are to be rearranged.

1. How many distinct arrangements are there?
2. How many of the arrangements start and end with the letter S ?
3. What is the probability that, in a randomly chosen arrangement, the S's are all together?

6 A volleyball squad has 11 players. A volleyball team consists of 6 players.

1. Find the total number of different teams that could be chosen from the squad. The squad has 5 women and 6 men.
2. Find the total number of different teams that contain at least 3 women. The squad includes a man and a woman who are married to one another.
3. It is given that the team chosen has exactly 3 women and all such teams are equally likely to be chosen. Calculate the probability that a team chosen includes the married couple.

2

1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space at least 15 times. Justify the use of the normal approximation in this case.
2. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).

12 A faulty random number generator generates odd digits according to the probability distribution for the random variable \(X\) given in the following table.

1. Find \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\).
3. Deduce the value of \(\operatorname { Var } ( X )\). A second random number generator generates odd digits each with equal probability. Both random generators are operated once.
4. Find the probability that both generate a prime number.
5. Given that the first generates 1, 3 or 5, find the probability that both generate a power of 3 . 1315 pupils, including two sisters, are placed in random order in a line.
6. What is the probability that the sisters are not next to each other?
7. How many arrangements are there with 9 pupils between the sisters? A team of 5 is chosen from the 15 pupils.
8. How many ways are there of choosing the team if no more than one of the sisters can be in the team? Having chosen the first team, a second team of 5 pupils is chosen from the remaining 10 pupils.
9. How many ways are there of choosing the teams if each sister is in one or other of the teams?

1. Find the number of different arrangements of the 9 letters in the word DELIVERED in which the three Es are together and the two Ds are not next to each other. [4]
2. Find the probability that a randomly chosen arrangement of the 9 letters in the word DELIVERED has exactly 4 letters between the two Ds. [5]

Five letters are selected from the 9 letters in the word DELIVERED.

1. Find the number of different selections if the 5 letters include at least one D and at least one E. [3]

The digits of the number 1223678 can be rearranged to give many different 7-digit numbers. Find how many different 7-digit numbers can be made if

1. there are no restrictions on the order of the digits, [2]
2. the digits 1, 3, 7 (in any order) are next to each other, [3]
3. these 7-digit numbers are even. [3]

Nine cards, each of a different colour, are to be arranged in a line.

1. How many different arrangements of the 9 cards are possible? [1]

The 9 cards include a pink card and a green card.

1. How many different arrangements do not have the pink card next to the green card? [3]

Consider all possible choices of 3 cards from the 9 cards with the 3 cards being arranged in a line.

1. How many different arrangements in total of 3 cards are possible? [2]
2. How many of the arrangements of 3 cards in part (iii) contain the pink card? [2]
3. How many of the arrangements of 3 cards in part (iii) do not have the pink card next to the green card? [2]

A box contains 5 discs, numbered 1, 2, 4, 6, 7. William takes 3 discs at random, without replacement, and notes the numbers on the discs.

1. Find the probability that the numbers on the 3 discs are two even numbers and one odd number. [3]

The smallest of the numbers on the 3 discs taken is denoted by the random variable \(S\).

1. By listing all possible selections (126, 246 and so on) draw up the probability distribution table for \(S\). [5]

Find the number of different ways that 6 boys and 4 girls can stand in a line if

1. all 6 boys stand next to each other, [3]
2. no girl stands next to another girl. [3]

There are 125 sixth-form students in a college, of whom 60 are studying only arts subjects, 40 only science subjects and the rest a mixture of both. Three students are selected at random, without replacement. Find the probability that

1. all three students are studying only arts subjects, [4]
2. exactly one of the three students is studying only science subjects. [3]

Among the families with two children in a large city, the probability that the elder child is a boy is \(\frac{5}{12}\) and the probability that the younger child is a boy is \(\frac{9}{16}\). The probability that the younger child is a girl, given that the elder child is a girl, is \(\frac{1}{4}\). One of the families is chosen at random. Using a tree diagram, or otherwise,

1. show that the probability that both children are boys is \(\frac{1}{8}\). [5 marks]

Find the probability that

1. one child is a boy and the other is a girl, [3 marks]
2. one child is a boy given that the other is a girl. [3 marks]

If three of the families are chosen at random,

1. find the probability that exactly two of the families have two boys. [3 marks]
2. State an assumption that you have made in answering part (d). [1 mark]