Set notation

In a nutshell

Events within a sample space or Venn diagram may be described using set notation. This will help you calculate certain probabilities.

Intersection

This is the set formed by the elements that belong to both events. It is written as $A\cap B$ .

Maths; Conditional probability; KS5 Year 13; Set notation

The probability is written as

$\boxed{P(A \cap B)}$

Note: If the events are independent, then $P(A\cap B)=P(A)\times P(B)$ .

Example 1

Find $P(A\cap B)$ in the following sample space:

Maths; Conditional probability; KS5 Year 13; Set notation

You have to divide the number of elements common to both sets by all the possible outcomes ( $9+7+4+1=21$ ):

$\underline{P(A\cap B)=\dfrac{1}{21}}$ 

Union

This is formed by all the elements from the events. It is written as $A\cup B$ .

Maths; Conditional probability; KS5 Year 13; Set notation

The probability is written as

$\boxed{P(A\cup B)}$

Note: If the events are mutually exclusive, then $P(A\cup B)=P(A)+P(B)$ .

Example 2

Consider the last Venn diagram. What is $P(A\cup B)$ ?

Find $A\cup B$ :

$A\cup B=7+4+1=12$ 

Divide this by the number of outcomes:

$\underline{P(A\cap B)=\dfrac{12}{21}=\dfrac{4}{7}}$ 

Complement

The complement refers to all the elements from the sample space that don't belong to the event. It is written as $A'$ and can also be called 'not $A$ '.

Maths; Conditional probability; KS5 Year 13; Set notation

The probability is written as

$\boxed{P(A') = 1-P(A)}$

Example 3

From the Venn diagram on the first example, what will $P(B')$ be?

There are $9+4=13$ elements that don't belong to $B'$ :

$\underline{P(B')=\dfrac{13}{21}}$