Table Of ContentBrighter Thinking
A Level Further
Mathematics for OCR A
Statistics Student Book (AS/ A Level)
Vesna Kadelburg, Ben Woolley, Paul Fannon and Stephen Ward
Contents
Introduction
How to use this resource
1 Counting principles and probability
Section 1: The product principle and the addition principle
Section 2: Permutations
Section 3: Combinations
Section 4: The exclusion principle
Section 5: Selections where order matters
Section 6: Keeping objects together or separated
Section 7: Permuting objects with repetitions
Section 8: Counting principles in probability
2 Discrete random variables
Section 1: Average and spread of a discrete random variable
Section 2: Expectation and variance of transformations of discrete random variables
Section 3: The discrete uniform distribution
Section 4: Binomial distribution
Section 5: The geometric distribution
3 Poisson distribution
Section 1: Using the Poisson model
4 Non-parametric hypothesis tests
Section 1: The single-sample sign test
Section 2: The single-sample Wilcoxon signed-rank test
Section 3: Matched-pairs (or paired-sample) tests
Section 4: Wilcoxon rank-sum test
Section 5: Normal approximations
Focus on … Proof 1
Focus on … Problem solving 1
Focus on … Modelling 1
Cross-topic review exercise 1
5 Correlation and regression
Section 1: Pearson’s product moment correlation coefficient
Section 2: Spearman’s rank correlation coefficient
Section 3: Linear regression
6 Chi-squared tests
Section 1: Contingency tables
Section 2: Yates’ correction
Section 3: Goodness-of-fit tests
7 Continuous distributions
Section 1: Continuous random variables
Section 2: Averages and measures of spread of continuous random variables
Section 3: Expectation and variance of functions of a random variable
Section 4: Cumulative distribution functions
Section 5: Piecewise-defined probability density functions
Section 6: Continuous uniform distribution
Section 7: Exponential distribution
Section 8: Distributions of functions of a continuous random variable
Section 9: Goodness-of-fit test with continuous distributions
8 Combining random variables
Section 1: Adding independent random variables
Section 2: Expectation and variance of the sample mean
Section 3: Unbiased estimates of the mean and the variance
Section 4: Linear combinations of normal variables
Section 5: The distribution of means of large samples
9 Further hypothesis tests and confidence intervals
Section 1: Hypothesis testing for the mean of a large sample
Section 2: Confidence intervals
Focus on … Proof 2
Focus on … Problem solving 2
Focus on … Modelling 2
Cross-topic review exercise 2
AS Level practice paper
A Level practice paper
Formulae
Answers
Worked solutions for chapter exercises
1 Counting principles and probability
2 Discrete random variables
3 Poisson distribution
4 Non-parametric hypothesis tests
5 Correlation and regression
6 Chi-squared tests
7 Continuous distributions
8 Combining random variables
9 Further hypothesis tests and confidence intervals
Worked solutions for cross-topic review exercises
Cross-topic review exercise 1
Cross-topic review exercise 2
AS Level practice paper
A Level practice paper
Glossary
Acknowledgements
Copyright
Introduction
You have probably been told that mathematics is very useful, yet it can often seem like a lot of techniques
that just have to be learned to answer examination questions. You are now getting to the point where you
will start to see where some of these techniques can be applied in solving real problems. However, as well
as seeing how maths can be useful we hope that anyone working through this book will realise that it can
also be incredibly frustrating, surprising and ultimately beautiful.
The book is woven around three key themes from the new curriculum:
Proof
Maths is valued because it trains you to think logically and communicate precisely. At a high level, maths is
far less concerned about answers and more about the clear communication of ideas. It is not about being
neat – although that might help! It is about creating a coherent argument that other people can easily
follow but find difficult to refute. Have you ever tried looking at your own work? If you cannot follow it
yourself it is unlikely anybody else will be able to understand it. In maths we communicate using a variety
of means – feel free to use combinations of diagrams, words and algebra to aid your argument. Once you
have attempted a proof, try presenting it to your peers. Look critically (but positively) at some other
people’s attempts. It is only through having your own attempts evaluated and trying to find flaws in other
proofs that you will develop sophisticated mathematical thinking. This is why we have included lots of
common errors in our ’work it out’ boxes – just in case your friends don’t make any mistakes!
Problem solving
Maths is valued because it trains you to look at situations in unusual, creative ways, to persevere and to
evaluate solutions along the way. We have been heavily influenced by a great mathematician and maths
educator George Polya, who believed that students were not just born with problem solving skills – they
were developed by seeing problems being solved and reflecting on their solutions before trying similar
problems. You may not realise it but good mathematicians spend most of their time being stuck. You need
to spend some time on problems you can’t do, trying out different possibilities. If after a while you have not
cracked it then look at the solution and try a similar problem. Don’t be disheartened if you cannot get it
immediately – in fact, the longer you spend puzzling over a problem the more you will learn from the
solution. You may never need to integrate a rational function in future, but we firmly believe that the
problem solving skills you will develop by trying it can be applied to many other situations.
Modelling
Maths is valued because it helps us solve real-world problems. However, maths describes ideal situations
and the real world is messy! Modelling is about deciding on the important features needed to describe the
essence of a situation and turning that into a mathematical form, then using it to make predictions,
compare to reality and possibly improve the model. In many situations the technical maths is actually the
easy part – especially with modern technology. Deciding which features of reality to include or ignore and
anticipating the consequences of these decisions is the hard part. Yet it is amazing how some fairly drastic
assumptions – such as pretending a car is a single point or that people’s votes are independent – can result
in models that are surprisingly accurate.
More than anything else, this book is about making links. Links between the different chapters, the topics
covered and the themes above, links to other subjects and links to the real world. We hope that you will
grow to see maths as one great complex but beautiful web of interlinking ideas.
Maths is about so much more than examinations, but we hope that if you take on board these ideas (and
do plenty of practice!) you will find maths examinations a much more approachable and possibly even
enjoyable experience. However, always remember that the results of what you write down in a few hours
by yourself in silence under exam conditions is not the only measure you should consider when judging
your mathematical ability – it is only one variable in a much more complicated mathematical model!
How to use this resource
Throughout this resource you will notice particular features that are designed to aid your learning. This
section provides a brief overview of these features.
In this chapter you will learn how to:
break down complicated questions into parts that are easier to count, and then combine them
together
count the number of ways to permute a set of objects
Learning objectives
A short summary of the content that you will learn in each chapter.
Before you start…
A Level You should know how to work
1 Evaluate
Mathematics with the factorial function.
Student Book 1,
Chapter 9
A Level You should know how to work 2 What is the probability of rolling a prime
Mathematics with basic probability. number on a fair dice?
Student Book 1,
Chapter 17
Before you start…
Points you should know from your previous learning and questions to check that you’re ready to start the
chapter.
WORKED EXAMPLE
The left-hand side shows you how to set out your working. The right-hand side explains the more
difficult steps and helps you understand why a particular method was chosen.
PROOF
Step-by-step walkthroughs of standard proofs and methods of proof.
WORK IT OUT
Can you identify the correct solution and find the mistakes in the two incorrect solutions?
Key point
A summary of the most important methods, facts and formulae.
Explore
Ideas for activities and investigations to extend your understanding of the topic.
Tip
Useful guidance, including ways of calculating or checking answers and using technology.
Each chapter ends with a Checklist of learning and understanding and a Mixed practice exercise,
which includes past paper questions marked with the icon .
In between chapters, you will find extra sections that bring together topics in a more synoptic way.
Focus on…
Unique sections relating to the preceding chapters that develop your skills in proof, problem solving and
modelling.
CROSS-TOPIC REVIEW EXERCISE
Questions covering topics from across the preceding chapters, testing your ability to apply what you
have learned.
You will find practice paper questions towards the end of the resource, as well as a glossary of key
terms (picked out in colour within the chapters), answers and full worked solutions.
Maths is all about making links, which is why throughout this book you will find signposts emphasising
connections between different topics, applications and suggestions for further research.
Rewind
Reminders of where to find useful information from earlier in your study.
Fast forward
Links to topics that you may cover in greater detail later in your study.
Focus on…
Links to problem-solving, modelling or proof exercises that relate to the topic currently being
studied.
Did you know?
Interesting or historical information and links with other subjects to improve your awareness
about how mathematics contributes to society.
Colour-coding of exercises
The questions in the exercises are designed to provide careful progression, ranging from basic fluency to
practice questions. They are uniquely colour-coded, as shown here.
1 Evaluate:
a i
ii
b i
ii
c i
ii
2 Find the number of arrangements of the letters ALGEBRA.
3 Find the number of permutations of the letters STATISTICIAN.
8 and Find
10 Find the number of ordered subsets of of size four.
11 Explain why the number of arrangements of copies of object A and y copies of object B is
Black – drill questions. These come in several parts, each with subparts i and ii. You only need attempt
subpart i at first; subpart ii is essentially the same question, which you can use for further practice if you
got part i wrong, for homework, or when you revisit the exercise during revision.
Green – practice questions at a basic level.
Blue – practice questions at an intermediate level.
Red – practice questions at an advanced level.
Purple – challenging questions that apply the concept of the current chapter across other areas of
maths.
Yellow – designed to encourage reflection and discussion.
– indicates content that is for A Level students only
– indicates content that is for AS Level students only.
– indicates a question that requires a calculator.
1 Counting principles and probability
In this chapter you will learn how to:
break down complicated questions into parts that are easier to count, and then
combine them together
count the number of ways to permute a set of objects
count the number of ways you can choose objects from a group
apply these tools to problems involving probabilities.
Before you start…
A Level Mathematics You should know how to work with 1 Evaluate
Student Book 1, Chapter 9 the factorial function.
A Level Mathematics You should know how to work with 2 What is the probability of rolling
Student Book 1, Chapter basic probability. a prime number on a fair dice?
17
Making maths count
Counting is the one of the first things you learn in Mathematics and at first it seems very simple. If you
were asked to count how many people there are in your school, this would not be too tricky. If you were
asked how many chess matches need to be played if everyone is to play everyone else, this is a little more
complicated. If you were asked how many different football teams could be chosen, you might find that the
numbers become far too large to count without using some mathematical techniques.
One of the main uses of this type of counting is in calculating probabilities. You already have tools such as
tree diagrams and Venn diagrams, but some problems are easier to solve by counting all the possibilities.
The methods can be applied to fields as diverse as games of chance, genetics and cryptography.
Although in the exam all the questions will be set in the context of probability, you need to learn about
various counting techniques first, before applying them to probability problems in Section 8.
Section 1: The product principle and the addition principle
Counting very small groups is simple. You need to break down more complicated problems into counting
small groups. But how do you combine these together to come up with an answer to the overall problem?
The answer lies in the product principle and the addition principle, which can be demonstrated by
considering order choices from the menu shown.
Mains
Pizza
Hamburger
Paella
Desserts
Ice cream
Fruit salad
Anna would like to order a main course and a dessert. She could make six different orders – three choices
for the main course and for each choice she makes there, two choices for dessert, so she multiplies the
individual possibilities.
Bob would like to order either a main course or a dessert. He could make five different orders – one of the
three main courses or one of the two desserts so he adds the individual possibilities.
You can use the notation to mean the number of ways of making a choice about
The product principle tells us that when you wish to select one option from
AND one option from you multiply the individual possibilities together.
Key point 1.1
Product principle:
The addition principle tells us that when you wish to select one option from or one option from you add
the individual possibilities together.
The addition principle has one caveat. You can only use it if there is no overlap between the choices for
and the choices for For example, you cannot apply the addition principle to counting the number of
ways of getting an odd number or a prime number on a dice. If there is no overlap between the choices for
and for the two events are mutually exclusive.
Key point 1.2
Addition principle:
if and are mutually exclusive.
The hardest part of applying the addition or product principles is breaking the problem down into relevant
parts. It is often useful to rewrite questions to emphasise whether what is required is ‘AND’ or ‘OR’.
WORKED EXAMPLE 1.1
WORKED EXAMPLE 1.1
An examination has ten questions in section A and four questions in section B. Calculate how many
different ways there are to choose questions if you must choose:
a one question from each section
b a question from either section A or section B.
a Choose one question from AND one from Describe the problem accurately.
‘AND’ means you should apply the product
principle.
b Choose one question from OR one from Describe the problem accurately.
‘OR’ means you should apply the addition
principle.
In the context of Worked example 1.1 you cannot have a repeat selection of an object. However, there are
situations where you might be able to do so.
WORKED EXAMPLE 1.2
In a class there is an award for best mathematician, best sportsperson and nicest person. People can
receive more than one award. In how many ways can the awards be distributed if there are twelve
people in the class?
Choose one of twelve people for the best mathematician Describe the problem accurately.
AND one of the twelve for best sportsperson
AND one of the twelve for nicest person
Apply the product principle.
EXERCISE 1A
1 If there are ways of doing , ways of doing and ways of doing , with , and mutually
exclusive, calculate how many ways there are of doing:
a i both and
ii both and
b i either or
ii either or
2 If there are four ways of doing , seven ways of doing and five ways of doing , with , and
mutually exclusive, calculate how many ways there are of doing:
a all of and
b exactly one of or
3 Show many different paths there are in this diagram:
a from
b from
c from
A B C D E
