Probability is a fundamental concept in statistics, explored in Chapter 14 of Cambridge Maths NSW Stage 6 for Year 11 students. This chapter introduces basic probability concepts, including experiments with various outcomes and the significance of conditional probability. Key topics include set theory, Venn diagrams, and practical problems involving sports tryouts and dice rolls. Designed for students preparing for advanced mathematics, this chapter provides a systematic approach to solving probability problems, enhancing understanding of statistical principles. It is essential for mastering concepts required for higher-level mathematics and statistics courses.

Key Points

  • Introduces basic probability concepts and their applications in statistics
  • Explains set theory and Venn diagrams for solving probability problems
  • Includes practical examples with sports tryouts and dice rolls
  • Covers conditional probability and its significance in experiments
Leo
Author:Pender et al.
Edition:2025
59 pages
Language:English
Type:Textbook
Probability
Leo
Author:Pender et al.
Edition:2025
59 pages
Language:English
Type:Textbook
Probability
145
/ 59
14
Probability
Chapter introduction
Probability arises when one performs an experiment that has various possible outcomes, but there is insufficient
information to predict which of these outcomes will occur. The classic examples of this are tossing a coin,
throwing a die, and drawing a card from a pack. Probability, however, is involved in almost every experiment
done in science, and is fundamental to understanding statistics.
This chapter reviews some basic ideas of probability and develops a more systematic approach to solving
probability problems. It concludes with the new topic of conditional probability.
CambridgeMATHS NSW Stage 6 – Mathematics Extension 1 ISBN 978-1-009-65480-7
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544 Chapter 14 Probability
14A
14A
Sets and Venn diagrams
Learning intentions
• Use the basic language and notation of set theory, with Venn diagrams.
• Relate the words ‘and’, ‘or’, and ‘not’ to intersection, union, and complement.
A probability experiment has many possible outcomes. We shall be using the language of set theory to handle
and keep track of this set of possible outcomes while calculating probabilities. A brief review of sets and Venn
diagrams is needed in preparation for their application to situations involving probability. The three key ideas
needed in probability are the intersection of sets, the union of sets, and the complement of a set.
Logic is very close to the surface when we talk about sets and Venn diagrams. The three ideas of intersection,
union and complement mentioned above correspond very precisely to the words ‘and’, ‘or’, and ‘not’.
Listing sets and describing sets
A set is a collection of things called elements or members. When a set is specified, it needs to be made absolutely
clear what things are its elements. This can be done by listing the elements inside curly brackets. For example,
S = {1, 3, 5, 7, 9},
which is read as ‘S is the set whose elements are 1, 3, 5, 7, and 9’.
A set can also be specified by writing a description of the elements inside curly brackets. For example,
T = {odd whole numbers less than 10},
which is read as ‘T is the set of odd whole numbers less than 10’.
Equal sets
Two sets are called equal if they have exactly the same elements. Hence the sets S and T in the previous
paragraph are equal — written as S = T . The order in which the elements are written doesn’t matter at all, and
neither does repetition. For example,
{1, 3, 5, 7, 9} = {3, 9, 7, 5, 1} = {5, 9, 1, 3, 7} = {1, 3, 1, 5, 1, 7, 9}.
The size of a set
A set may be finite, such as the set above of odd whole numbers less than 10, or infinite, such as the set of all
whole numbers.
If a set S is finite, then the symbol |S | is used to mean the number of elements of S . For example:
If A = {5, 6, 7, 8, 9, 10}, then |A| = 6.
If B = {letters in the alphabet}, then |B| = 26.
If C = {12}, then |C| = 1.
If D = {odd numbers between 1.5 and 2.5}, then |D| = 0.
The number of elements in a set S can also be written as n(S).
The empty set
The last set D above is called the empty set, because it has no elements at all. The usual symbol for the empty set
is ∅. There is only one empty set, because any two empty sets have exactly the same elements (that is, none at
all), and so are equal.
CambridgeMATHS NSW Stage 6 – Mathematics Extension 1 ISBN 978-1-009-65480-7
Year 11 Photocop
y
in
g
is restricted under law and this material must not be transferred to another part
y
.
© Pender et al. 2025 Cambrid
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e Universit
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Press & Assessment
14A Sets and Venn diagrams 545
1 Set terminology and notation
• A set is a collection of elements. For example,
S = {1, 3, 5, 7, 9} and T = {odd integers from 0 to 10}.
• Two sets are called equal if
they have exactly the same elements.
• The number of elements in a finite set S is
written as |S|.
• The empty
set is written as ∅ — there is only one empty set.
The number of elements in a set S can also be written as n(S).
Intersection and union, and disjoint sets
There are two obvious ways of combining two sets A and B.
• The intersection A ∩ B of A and B is the set of elements in A and in B.
• The union A ∪ B of A and B is the set of elements in A or in B.
For example, if A = {0, 1, 2, 3} and B = {1, 3, 6}, then
A ∩ B = {1, 3}
A ∪ B = {0, 1, 2, 3, 6}
• Two sets A and B are called disjoint if they have no elements in common, that is, if A ∩ B = ∅.
For example, if A = {2, 4, 6, 8} and B = {1, 3, 5, 7}, then
A ∩ B = ∅,
so A and B are disjoint.
‘And’ means intersection, ‘or’ means union
The important mathematical words ‘and’ and ‘or’ can be interpreted in terms of union and intersection:
A ∩ B = {elements that are in A and
in B}
A ∪ B = {elements that are in A or
in B}
Note: The word ‘or’ in mathematics always means ‘and/or’. For example, all the elements of A ∩ B are
members of A ∪ B.
2 The intersection and union of sets
Intersection of sets (corresponding to ‘and’):
• The intersection of
two sets A and B is the set of elements in A and in B:
A ∩ B = elements in A and
in B,
• The sets A and B are
called disjoint when A ∩ B = ∅.
Union of sets (corresponding to ‘or’):
• The union of A and B is
the set of elements in A or in B:
A ∪ B = elements in A or
in B.
Warning: The word ‘or’ in mathematics always means ‘and/or’.
CambridgeMATHS NSW Stage 6 – Mathematics Extension 1 ISBN 978-1-009-65480-7
Year 11 Photocop
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in
g
is restricted under law and this material must not be transferred to another part
y
.
© Pender et al. 2025 Cambrid
g
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Press & Assessment
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FAQs

What are the key concepts covered in Chapter 14 on probability?
Chapter 14 on probability covers essential concepts such as the definition of probability, the role of experiments with uncertain outcomes, and the importance of conditional probability. It introduces set theory and Venn diagrams as tools for visualizing and solving probability problems. The chapter also includes practical examples, such as calculating probabilities related to sports tryouts and rolling dice, which help students apply theoretical concepts to real-world scenarios.
How does this chapter approach the topic of conditional probability?
The chapter concludes with a focus on conditional probability, explaining its relevance in determining the likelihood of an event given that another event has occurred. It provides examples that illustrate how to calculate conditional probabilities and emphasizes the importance of understanding this concept in the context of statistical analysis. By mastering conditional probability, students can better analyze complex situations where outcomes are interdependent.
What types of problems are included in the probability chapter?
The probability chapter includes a variety of problems that require students to apply their understanding of probability concepts. Examples range from simple calculations, such as determining the probability of rolling a certain number on a die, to more complex scenarios involving multiple events, like sports tryouts. Students are encouraged to use Venn diagrams to visualize relationships between different sets, enhancing their problem-solving skills.
What is the significance of Venn diagrams in probability?
Venn diagrams are significant in probability as they provide a visual representation of the relationships between different sets. In this chapter, students learn how to use Venn diagrams to solve problems involving multiple events, such as those related to sports tryouts. By illustrating overlaps and exclusive outcomes, Venn diagrams help clarify complex probability scenarios and assist in calculating probabilities more effectively.
How does this chapter support Year 11 students in their mathematics studies?
This chapter supports Year 11 students by providing a comprehensive introduction to probability, which is a critical component of their mathematics curriculum. It offers a structured approach to understanding key concepts, practical applications, and problem-solving techniques. By engaging with the material, students build a solid foundation in probability that is essential for success in higher-level mathematics and statistics courses.

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