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Conditional probability

Set notation

Set notation

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Explainer Video

Tutor: Alice

Summary

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∩BA\cap BA∩B.


Maths; Conditional probability; KS5 Year 13; Set notation

The probability is written as 

​P(A∩B)\boxed{P(A \cap B)}P(A∩B)​​​


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


Example 1

Find P(A∩B)P(A\cap B)P(A∩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=219+7+4+1=219+7+4+1=21):


P(A∩B)=121‾\underline{P(A\cap B)=\dfrac{1}{21}}P(A∩B)=211​​​​


Union

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


Maths; Conditional probability; KS5 Year 13; Set notation

The probability is written as

​P(A∪B)\boxed{P(A\cup B)}P(A∪B)​​​


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


Example 2

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


Find A∪BA\cup BA∪B:

A∪B=7+4+1=12A\cup B=7+4+1=12A∪B=7+4+1=12​​


Divide this by the number of outcomes:

P(A∩B)=1221=47‾\underline{P(A\cap B)=\dfrac{12}{21}=\dfrac{4}{7}}P(A∩B)=2112​=74​​​​



Complement

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


Maths; Conditional probability; KS5 Year 13; Set notation

The probability is written as 

P(A′)=1−P(A)\boxed{P(A') = 1-P(A)}P(A′)=1−P(A)​​​


Example 3

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


There are 9+4=139+4=139+4=13 elements that don't belong to B′B'B′:

​P(B′)=1321‾\underline{P(B')=\dfrac{13}{21}}P(B′)=2113​​


​

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Learn with Basics

Learn the basics with theory units and practise what you learned with exercise sets!
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Venn diagrams

Unit 1

Venn diagrams

Sets and Venn diagrams

Unit 2

Sets and Venn diagrams

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Set notation

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FAQs - Frequently Asked Questions

What is the complement of a set?

It is all the elements in the sample space that don't belong to that set.

What is the union of two sets?

It is all the elements present on the sets.

What is the intersection of two sets?

It is all the elements that belong to both sets.

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