Cambridge International AS & A Level Mathematics: Probability & Statistics 1 Coursebook (Cambridge Assessment International Education)

266 Pages·2018·29.278 MB·English

by  Dean Chalmers, Julian Gilbey (editor)

Cambridge International
AS & A Level Mathematics:
Probability & Statistics 1
Coursebook

Copyright Material - Review Only - Not for Redistribution
Copyright Material - Review Only - Not for Redistribution
y
p
o
C

w
e
i
v
e
R

-

s
s
e
r
P

y
t
i
s
r
e

v y
i p
n
o
U
C

e
w
g
d e
ri vi
b e
m R
a
-
C Dean Chalmers
s
- s
e
y
Sreries Editor: Julian Gilbey
p P
o

C y
t
w siCambridge International
r
e e

vi v y
e nAi S & A Level Mpathematics:
R U o
C

e
w
Probag bility & Statistics 1
d e
ri vi
b e
m R
a
-
C Coursebook
s
- s
e
y
r
p P
o

C y
t
i
w s
r
e e

vi v y
e ni p
R U o
C

e
w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y
r
p P
o

C y
t
i
w s
r
e e

vi v y
e ni p
R U o
C

e
w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y Copyright Material - Review Only - Not for Redistribution
r
p P
o

C y
t
i
w s
r
e e
vi v
e ni
R U

e
g
d
i
r
b
m
a
C

-

y
p
o
C

w
e
i
v
e
R
y
p
o
C

w
e
i
v
e
R

-

s
s
e
r
P

y
t
i
s
r
e

v y
i p
n
o
U
C

e
w
g
d e
ri vi
University Printing House,b Cambridge CB2 8BS, United Kingdom e
m R
One Liberty Plaza, 20th Floor, New York, NY 10006, USA
a
477 WilliamstownC Road, Port Melbourne, VIC 3207, Australia -
s
314–321, 3rd F-l oor, Plot 3, Splendor Forum, Jasola District Centre, Nsew Delhi – 110025, India
79 Anson Ro ad, #06–04/06, Singapore 079906 e
y
r
p P
Camboridge University Press is part of the University of Cambridge.

C y
It furthers the University’s mission by disseminating knowledge in the pursuit of
t
we ducation, learning and research at the highest internastiional levels of excellence.
r
e e
www.cambridge.org
vi v y
e Information on this title: www.cambridge.orngi/9781108407304 p
R © Cambridge University Press 2018 U o
C
This publication is in copyright. Subjec t to statutory exception
e
and to the provisions of relevant collective licensing agreements, w
g
no reproduction of any part mayd take place without the written e
permission of Cambridge Unriiversity Press. vi
First published 2018 b e
m R
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
a
Printed in the United Kingdom by Latimer Trend -
C
s
A catalogue re-c ord for this publication is available from the British Librsary
ISBN 978-1-1 08-40730-4 Paperback e
y
r
Cambridpge University Press has no responsibility for the persisPtence or accuracy
of URoLs for external or third-party internet websites referre d to in this publication,
andC does not guarantee that any content on such websitesy is, or will remain,
t
wa ccurate or appropriate. Information regarding pricess, itravel timetables, and other
factual information given in this work is correct at trhe time of first printing but
e e
Cambridge University Press does not guarantee the accuracy of such information thereafter.
vi v y
e ® IGCSE is a registered trademark ni p
R Past exam paper questions throughout areU reproduced by permission o
C
of Cambridge Assessment Internationa l Education. Cambridge Assessment
e
International Education bears no responsibility for the example answers to questions w
g
taken from its past question papers which are contained in this publication.
d e
The questions, example answerris, marks awarded and/or comments that appear in this bvoiok were
written by the author(s). In ebxamination, the way marks would be awarded to answeres like these
may be different. m R
a
-
NOTICE TO TEACCHERS IN THE UK
s
It is illegal to -re produce any part of this work in material form (includsing
photocopyin g and electronic storage) except under the following cirecumstances:
y
(i) w herpe you are abiding by a licence granted to your school oPr rinstitution by the
Coopyright Licensing Agency;

(ii) C where no such licence exists, or where you wish to exceyed the terms of a licence,
t
and you have gained the written permission of Camibridge University Press;
w s
(iii) w here you are allowed to reproduce without permission under the provisions
r
e of Chapter 3 of the Copyright, Designs and Paetents Act 1988, which covers, for

vi example, the reproduction of short passagesv within certain types of educational y
e anthology and reproduction for the purpnoises of setting examination questions. p
R U o
C

e
w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y Copyright Material - Review Only - Not for Redistribution
r
p P
o

C y
t
i
w s
r
e e
vi v
e ni
R U

e
g
d
i
r
b
m
a
C

-

y
p
o
C

w
e
i
v
e
R
y
p
o
C

w
e
i
v
e
R

-

s
s
e
r
P

y
t
i
s
r
e Contents
v y
i p
n
o
U
C

e
Contents w
g
d e
ri vi
b e
Series introdumction R vi
a
-
C
How to use this book s viii
- s
e
y
r
Ackpnowledgements P x
o

C y
t
w 1 Representation of data si 1
r
e e
1.1 Types of data 2
vi v y
e 1.2 Representation of discrete dnaita: stem-and-leaf diagrams p 3
R U o
1.3 Representation of continuous data: histograms C 6

e
1.4 Representation of cogntinuous data: cumulative frequency graphsw 13
d e
1.5 Comparing diffreirent data representations vi 20
b e
End-of-chapter review exercise 1 24
m R
a
-
2 MeasurCes of central tendency 26
s
- s
2.1 Th e mode and the modal class e 28
y
r
2.2 pThe mean P 30
o
iii
2C.3 The median y 42
t
i
w End-of-chapter review exercise 2 s 51
r
e e

vi v y
e 3 Measures of variation ni p 54
R U o
3.1 The range C 55

e
3.2 The interquartile range and percentiles w 56
g
d e
3.3 Variance and standard deviation 65
ri vi
End-of-chapter revbiew exercise 3 e 83
m R
a
-
Cross-topCic review exercise 1 86
s
- s
e
4 Pryobability 90
r
p P
o
4.1 Experiments, events and outcomes 91
C y
t
4.2 Mutually exclusive events and the adidition law 94
w s
r
e 4.3 Independent events and the multeiplication law 100

vi v y
e 4.4 Conditional probability ni p 108
R U o
4.5 Dependent events and conditional probability C 112

e
End-of-chapter review exercise 4 w 118
g
d e
5 Permutations raind combinations vi 122
b e
m R
5.1 The factorial function 124
a
-
5.2 PermutCations 125
s
- s
e
y Copyright Material - Review Only - Not for Redistribution
r
p P
o

C y
t
i
w s
r
e e
vi v
e ni
R U

e
g
d
i
r
b
m
a
C

-

y
p
o
C

w
e
i
v
e
R
y
p
o
C

w
e
i
v
e
R

-

s
s
e
r
P

y
t
i
s
r
Cambridge International AS & A Levele Mathematics: Probability & Statistics 1
v y
i p
n
o
U
C

e
5.3 Combinations g w 135
d e
5.4 Problem solving with permutations and combinations 138
ri vi
End-of-chapter reviewb exercise 5 e 143
m R
a
-
Cross-topic Creview exercise 2 146
s
- s
e
6 Probyability distributions r 149
p P
o
6.1 Discrete random variables 150
C y
t
6 .2 Probability distributions i 150
w s
r
e 6.3 Expectation and variance of a discreete random variable 156

vi v y
e End-of-chapter review exercise 6 ni p 162
R U o
C
7 The binomial and geom etric distributions 165
e
w
g
7.1 The binomial distribdution e 166
7.2 The geometric disrtiribution vi 175
b e
m R
End-of-chapter review exercise 7 185
a
-
C
s
8 The no-r mal distribution s 187
e
8.1 Conytinuous random variables r 188
p P
iv 8.2 oThe normal distribution 193
C y
8 .3 Modelling with the normal distributioni t 205
w s
r
e 8.4 The normal approximation to the beinomial distribution 208

vi v y
e End-of-chapter review exercise 8 ni p 215
R U o
C
Cross-topic review exerci se 3 217
e
w
g
d e
Practice exam-stylrei paper vi 220
b e
m R
The standard normal distribution function 222
a
-
C
s
Answers - s 223
e
y
r
p P
Glososary 245

C y
t
i
w s
Index 247
r
e e

vi v y
e ni p
R U o
C

e
w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y Copyright Material - Review Only - Not for Redistribution
r
p P
o

C y
t
i
w s
r
e e
vi v
e ni
R U

e
g
d
i
r
b
m
a
C

-

y
p
o
C

w
e
i
v
e
R
y
p
o
C

w
e
i
v
e
R

-

s
s
e
r
P

y
t
i
s
r
e

v y
i p
n
o
U
C

e
w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y
r
p P
o

C y
t
i
w s
r
e e

vi v y
e ni p
R U o
C

e
w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y
r
p P
o

C y
t
i
w s
r
e e

vi v y
e ni p
R U o
C

e
w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y
r
p P
o

C y
t
i
w s
r
e e

vi v y
e ni p
R U o
C

e
w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y Copyright Material - Review Only - Not for Redistribution
r
p P
o

C y
t
i
w s
r
e e
vi v
e ni
R U

e
g
d
i
r
b
m
a
C

-

y
p
o
C

w
e
i
v
e
R
y
p
o
C

w
e
i
v
e
R

-

s
s
e
r
P

y
t
i
s
r
Cambridge International AS & A Levele Mathematics: Probability & Statistics 1
v y
i p
n
o
U
C

e
Series introduction w
g
d e
ri vi
b e
m R
Cambridge International AS & A Level Mathematics can be a life-changing course. On the one hand, it is a
a
-
facilitating subCject: there are many university courses that ei ther require an A Level or equivalent qualification in
s
mathematic-s or prefer applicants who have it. On the othser hand, it will help you to learn to think more precisely
e
and logicyally, while also encouraging creativity. Doing mathematics can be like doing art: just as an artist needs to
r
p P
master her tools (use of the paintbrush, for example) and understand theoretical ideas (perspective, colour wheels
o

andC so on), so does a mathematician (using toolsy such as algebra and calculus, which you will learn about in this
t
wc ourse). But this is only the technical side: thsei joy in art comes through creativity, when the artist uses her tools
e to express ideas in novel ways. Mathemateicrs is very similar: the tools are needed, but the deep joy in the subject

vi comes through solving problems. v y
e ni p
R U o
You might wonder what a mathematical ‘problem’ is. This is a very good questCion, and many people have offered

different answers. You might lieke to write down your own thoughts on thisw q uestion, and reflect on how they
g
change as you progress thrdough this course. One possible idea is that a meathematical problem is a mathematical
question that you do norti immediately know how to answer. (If you dovi know how to answer it immediately, then
b e
we might call it an ‘exercise’ instead.) Such a problem will take time to answer: you may have to try different
m R
approaches, using different tools or ideas, on your own or with others, until you finally discover a way into it. This
a
-
may take minuCtes, hours, days or weeks to achieve, and your sense of achievement may well grow with the effort it
s
has taken. - s
e
y
r
p P
In addition to the mathematical tools that you will learn in this course, the problem-solving skills that you
o
vi
wilCl develop will also help you throughout life, wyhatever you end up doing. It is very common to be faced with
t
wp roblems, be it in science, engineering, mathseimatics, accountancy, law or beyond, and having the confidence to
e systematically work your way through theemr will be very useful.

vi v y
R e This series of Cambridge InternatiUonnail AS & A Level Mathematics coursebookso, wpritten for the Cambridge
Assessment International Education syllabus for examination from 2020, will sCupport you both to learn the

mathematics required for thesee examinations and to develop your mathemwa tical problem-solving skills. The new
g
examinations may well incdlude more unfamiliar questions than in the paest, and having these skills will allow you
to approach such questiroins with curiosity and confidence. vi
b e
m R
In addition to problem solving, there are two other key concepts that Cambridge Assessment International
a
-
Education haveC introduced in this syllabus: namely commun ication and mathematical modelling. These appear
s
in various f-o rms throughout the coursebooks. s
e
y
r
p P
Communication in speech, writing and drawing lies at the heart of what it is to be human, and this is no less
o

truCe in mathematics. While there is a temptationy to think of mathematics as only existing in a dry, written form
t
wi n textbooks, nothing could be further from stihe truth: mathematical communication comes in many forms, and
r
e discussing mathematical ideas with colleaegues is a major part of every mathematician’s working life. As you study

vi this course, you will work on many probvlems. Exploring them or struggling with themy together with a classmate
R e will help you both to develop your Uunniderstanding and thinking, as well as improvoinpg your (mathematical)
communication skills. And being able to convince someone that your reasoninCg is correct, initially verbally and

then in writing, forms the hearet of the mathematical skill of ‘proof’. w
g
d e
ri vi
b e
m R
a
-
C
s
- s
e
y Copyright Material - Review Only - Not for Redistribution
r
p P
o

C y
t
i
w s
r
e e
vi v
e ni
R U

e
g
d
i
r
b
m
a
C

-

y
p
o
C

w
e
i
v
e
R
y
p
o
C

w
e
i
v
e
R

-

s
s
e
r
P

y
t
i
s
r
e Series introduction
v y
i p
n
o
U
C

e
Mathematical modelling igs where mathematics meets the ‘real world’. Thwere are many situations where people need
to make predictions or tdo understand what is happening in the world,e and mathematics frequently provides tools
to assist with this. Marithematicians will look at the real world situavtiion and attempt to capture the key aspects
b e
of it in the form omf equations, thereby building a model of realiRty. They will use this model to make predictions,
and where posasible test these against reality. If necessary, th ey will then attempt to improve the model in order
-
to make bettCer predictions. Examples include weather prse diction and climate change modelling, forensic science
(to under-s tand what happened at an accident or crime sscene), modelling population change in the human, animal
e
and plyant kingdoms, modelling aircraft and ship brehaviour, modelling financial markets and many others. In this
p P
coourse, we will be developing tools which are vital for modelling many of these situations.

C y
t
i
w To support you in your learning, these cousrsebooks have a variety of new features, for example:
r
e e

vi ■ Explore activities: These activities avre designed to offer problems for classroom usey. They require thought and
e ni p
R deliberation: some introduce aU new idea, others will extend your thinking, whiole others can support consolidation.
C
The activities are often best approached by working in small groups and then sharing your ideas with each other
e
and the class, as they areg not generally routine in nature. This is one of twhe ways in which you can develop problem-
solving skills and confiddence in handling unfamiliar questions. e
■ Questions labelledb rais P , M or PS: These are questions with a epavirticular emphasis on ‘Proof’, ‘Modelling’ or
‘Problem solvinmg’. They are designed to support you in preparRing for the new style of examination. They may or
may not be haarder than other questions in the exercise.
-
C
■ The language of the explanatory sections makes much msore use of the words ‘we’, ‘us’ and ‘our’ than in previous
cour se-b ooks. This language invites and encourages eyosu to be an active participant rather than an observer, simply
y
fopllowing instructions (‘you do this, then you doP trhat’). It is also the way that professional mathematicians usually
owrite about mathematics. The new examinatio ns may well present you with unfamiliar questions, and if you are vii
C y
used to being active in your mathematics, ytou will stand a better chance of being able to successfully handle such
i
w challenges. s
r
e e

vi v y
e At various points in the books, thenire are also web links to relevant Undergroundp Mathematics resources,
R which can be found on the free uUndergroundmathematics.org website. Undergroound Mathematics has the aim
C
of producing engaging, rich m aterials for all students of Cambridge International AS & A Level Mathematics
e
w
and similar qualifications.g These high-quality resources have the potential to simultaneously develop your
d e
mathematical thinking skills and your fluency in techniques, so we do encourage you to make good use of them.
ri vi
b e
We wish you everym success as you embark on this course. R
a
-
C
Julian Gilbey s
- s
London , 2018 e
y
r
p P
o

C y
t
i
w s
r
e e

vi v y
e ni p
R U o
C
Past exam paper questions throu ghout are reproduced by permission of Cambridge Assessment International Education.
e
w
Cambridge Assessment Intergnational Education bears no responsibility for the example answers to questions taken from its
d e
past question papers which are contained in this publication.
ri vi
b e
The questions, exammple answers, marks awarded and/or comments thatR appear in this book were written by the author(s). In
examination, thea way marks would be awarded to answers like thes-e may be different.
C
s
- s
e
y Copyright Material - Review Only - Not for Redistribution
r
p P
o

C y
t
i
w s
r
e e
vi v
e ni
R U

e
g
d
i
r
b
m
a
C

-

y
p
o
C

w
e
i
v
e
R
y
p
o
C

w
e
i
v
e
R

-

s
s
e
r
P

y
t
i
s
r
e

v y
i p
n
S c U p o q
How to use this book C

e
w
g
d e
p riq vi
b e
m R
Throughout this book you will notice particular features that are designed to help your learning.
a
-
This section proCvides a brief overview of these features.
s
- s
y e The
r
p P
I■n this ochapter you will learn how to: PREREQUISITE KNOWLEDGE
■ dCisplay numerical data in stem-and-leaf diagrams, histograms and cumulative frtequyency graphs Where it comes from What you should be able to do Check your skills
e w ■ isnetleecrtp raent astpaptirsotpicraial tdea mtae pthreosde nfoterd d iinsp vlaayriionug sd faotram.s ersi IMGaCthSeEm® a/t Oic sLevel Olsoimbwtpearli enb p oaruponpbdrloes mptorsi aswotehl ueutnpio pgneivsr e oanfn d 1 taAo rtiehtcset laneneaagsrut elpsaotr s mpsilebotltre em .p Feearinismudr:eetse r2s0mh abyp 1e2m, both
data to a specified accurac y.
vi v y b the upper boundary of its area.
e Learning objectives indicate the imponritant IMGaCthSeEm / aOti cLsevel Chiostnosgtrruacmt sa wndit hin eteqrupparle at nd 2 Atw hoi cstlaosgsreasm o fi sd dartaaw. Tnh teo croelpurmesne n t
R U unequal intervaols. widths are 3cm and 4cm, and the column heights
concepts within each chapter and help you to C are 8cm and 6cm, respectively. What do we know
about the frequencies of these two classes?
navigate through the coursebooke. IGCSE / O Level Cowns truct and use cumulative 3 The heights of 50 trees are measured: 17 trees
g Mathematics frequency diagrams. are less than 3m; 44 trees are less than 4m; and
d e all of the trees are less than 5m. Determine, by
drawing a cumulative frequency diagram, how
ri vi many trees have heights:
b e a between 3 and 4m
m R b of 4m or more.
KEY POINT 1.2
a
-
D ata in a stemC-and- Key point boxes contain s PW rerequisite knowledge exercises identify prior learning
leaf diag-ra m are a summary of the most s that you need to have covered before starting the chapter.
e
ordereyd in rows of important methods, facts r Try the questions to identify any areas that you need to
eqouapl widths. and formulae. P review before continuing with the chapter.
viii
C y
t
i
w s WORKED EXAMPLE 5.14
r
e e
How many distinct three-digit numbers can be made from five cards, each with one
vi A cumulative frequency graph c v of the digits 5,5,7,8 and 9 written on iyt?
e ni p
R U Answer li o
Key terms are important terms in the topic that you The 55 is a repeated digit, so wCe must investigate three situations separately.

are learning. They are highlightgede in orange bold. cf nNuom 5bse sresl.ected: 3P3=6w t hree-digit Tarhrea ndgigeidt.s 7,8 and 9 are selected and
The glossary contains clear definitions of these key terms.
r all i rid tOhnreee -55d isgeilte c ntvuemid:b 3eCers2.×3!=18 sTewleoc tdeidg iatsn dfr oarmra 7n,g8e adn wd i9th a are 5 .
b e
EXPLORE 3.5 a m tTh wreoe 5-Rds isgeilte n cutemd:b 3eCrs1.×23!!=9 Oannde adrirgaitn fgreodm w 7it,h8 tawnod 59s i.s selected
-6+18+9=33 three-digit numbers can
C s be made.
The following table shows three students’ marks out of 20 in the same five tests.
- s
1st 2nd 3rd 4th 5th e
y
Amber p 12 17 11 9 16 x Pr Worked examples provide step-by-step approaches to
Butio 11 16 10 8 15 x–1
answering questions. The left side shows a fully worked
CChen 15 20 14 12 19 x+3 y
t solution, while the right side contains a commentary
Note that Buti’s marks are consistently 1 less than Amber’s and that Chein’s marks are
w s
consistently 3 more than Amber’s. This is indicated in the last columrn of the table. explaining each step in the working.
e For each student, calculate the variance and standard deviation.e
vi Can you explain your results, and do they apply equally to thev range and y
e interquartile range? ni p
R U TIP o
C

e A variable is d enoted
Explore boxes contain enrichment activities for extension w
g
work. These activities promodte group-work and peer- by an uppeer-case letter Tip boxes contain helpful
to-peer discussion, and arrei intended to deepen your and itsv ipossible values guidance about calculating
b x by thee same lower-case or checking your answers.
understanding of a comncept. (Answers to+ the Explore leRtter.
questions are provaided in the Teacher’s Resource.)
-
C
s
- s
e
y Copyright Material - Review Only - Not for Redistribution
r
p P
o

C y
t
i
w s
r
e e
vi v
e ni
R U

e
g
d
i
r
b
m
a
C

-

y
p
o
C

w
e
i
v
e
R