Mathematics Learning Centre
Basic concepts in probability
Sue Gordon
2005
c University of Sydney
,Mathematics Learning Centre, University of Sydney 1
1 Set Notation
You may omit this section if you are familiar with these concepts.
A set is a collection of objects.
We often specify a set by listing its members, or elements, in parentheses like this {}.
For example A = {2, 4, 6, 8} means that A is the set consisting of numbers 2,4,6,8.
We could also write A ={even numbers less than 9}.
The union of A and B is the set of elements which belong to A or to B (or both) and
can be written as A ∪ B.
The intersection of A and B is the set of elements which belong to both A and B, and
can be written as A ∩ B.
The complement of A, frequently denoted by A, is the set of all elements which do not
belong to A. In making this definition we assume that all elements we are thinking about
belong to some larger set U , which we call the universal set.
The empty set, written ∅ or {}, means the set with no elements in it.
A set C is a subset of A if all the elements in C are also in A.
For example, let
U = {all positive numbers ≤ 10}
A = {2, 4, 6, 8}
B = {1, 2, 3}
C = {6, 8}
Sets A, B and U may be represented in a Venn Diagram as follows:
5
A B
7
4 1 U
2 9
6
8 3 10
, Mathematics Learning Centre, University of Sydney 2
A intersection B, A ∩ B, is shown in the Venn diagram by the overlap of the sets A and
B, A ∩ B = {2}.
The union of the sets A and B, A ∪ B, is the set of elements that are in A = {2, 4, 6, 8}
together with the elements that are in B = {1, 2, 3} including each element once only.
So, A ∪ B = {1, 2, 3, 4, 6, 8}.
The complement of A is the set A is contains all the elements in U which are not in A.
So, A = {1, 3, 5, 7, 9, 10}.
C is a subset of A as every element in C = {6, 8} is also in A = {2, 4, 6, 8}.
2 Finite Equiprobable Spaces
In loose terms, we say that the probability of something happening is 14 , if, when the
experiment is repeated often under the same conditions, the stated result occurs 25% of
the time.
For the moment, we will confine our discussion to cases where there are a finite number
of equally likely outcomes.
For example, if a coin is tossed, there are two equally likely outcomes: heads (H) or tails
(T). If a die is tossed, there are six equally likely outcomes: 1,2,3,4,5,6.
Some Notation
The set of all possible outcomes of the given experiment is called the sample space. An
event is a subset of a sample space.
Calculating Probabilities
Look again at the example of rolling a six faced die. The possible outcomes in this
experiment are 1,2,3,4,5,6, so the sample space is the set {1,2,3,4,5,6}. The ‘event’ of
‘getting a 6’ is the subset {6}. We represent this in the following diagram.
Sample space
1 2 3
4 5 6 Event
Basic concepts in probability
Sue Gordon
2005
c University of Sydney
,Mathematics Learning Centre, University of Sydney 1
1 Set Notation
You may omit this section if you are familiar with these concepts.
A set is a collection of objects.
We often specify a set by listing its members, or elements, in parentheses like this {}.
For example A = {2, 4, 6, 8} means that A is the set consisting of numbers 2,4,6,8.
We could also write A ={even numbers less than 9}.
The union of A and B is the set of elements which belong to A or to B (or both) and
can be written as A ∪ B.
The intersection of A and B is the set of elements which belong to both A and B, and
can be written as A ∩ B.
The complement of A, frequently denoted by A, is the set of all elements which do not
belong to A. In making this definition we assume that all elements we are thinking about
belong to some larger set U , which we call the universal set.
The empty set, written ∅ or {}, means the set with no elements in it.
A set C is a subset of A if all the elements in C are also in A.
For example, let
U = {all positive numbers ≤ 10}
A = {2, 4, 6, 8}
B = {1, 2, 3}
C = {6, 8}
Sets A, B and U may be represented in a Venn Diagram as follows:
5
A B
7
4 1 U
2 9
6
8 3 10
, Mathematics Learning Centre, University of Sydney 2
A intersection B, A ∩ B, is shown in the Venn diagram by the overlap of the sets A and
B, A ∩ B = {2}.
The union of the sets A and B, A ∪ B, is the set of elements that are in A = {2, 4, 6, 8}
together with the elements that are in B = {1, 2, 3} including each element once only.
So, A ∪ B = {1, 2, 3, 4, 6, 8}.
The complement of A is the set A is contains all the elements in U which are not in A.
So, A = {1, 3, 5, 7, 9, 10}.
C is a subset of A as every element in C = {6, 8} is also in A = {2, 4, 6, 8}.
2 Finite Equiprobable Spaces
In loose terms, we say that the probability of something happening is 14 , if, when the
experiment is repeated often under the same conditions, the stated result occurs 25% of
the time.
For the moment, we will confine our discussion to cases where there are a finite number
of equally likely outcomes.
For example, if a coin is tossed, there are two equally likely outcomes: heads (H) or tails
(T). If a die is tossed, there are six equally likely outcomes: 1,2,3,4,5,6.
Some Notation
The set of all possible outcomes of the given experiment is called the sample space. An
event is a subset of a sample space.
Calculating Probabilities
Look again at the example of rolling a six faced die. The possible outcomes in this
experiment are 1,2,3,4,5,6, so the sample space is the set {1,2,3,4,5,6}. The ‘event’ of
‘getting a 6’ is the subset {6}. We represent this in the following diagram.
Sample space
1 2 3
4 5 6 Event
