Probability and Venn Diagrams

Probability and Venn Diagrams

A LevelAQAEdexcelOCRAQA 2022Edexcel 2022OCR 2022

Probability and Venn Diagrams

Probability is a measure of how likely something is to happen.

It always falls between \mathbf{0} and \mathbf{1}, with 0 being impossible and 1 being certain.

Notation: The probability of an event A is \mathbb{P}(A).

A Level AQA Edexcel OCR

Calculating Probabilities

To calculate the probability of an event, the formula is:

\dfrac{\color{red}\text{number of desired outcomes}}{\color{blue}\text{total number of possible outcomes}}

Example: What is the probability of obtaining an even number when rolling a 6-sided die?

Our possible outcomes are \color{blue}1,2,3,4,5,6, of which \color{red}2,4,6 are even. This gives \color{red}3 desired outcomes, out of \color{blue}6 possible outcomes, so:

\mathbb{P}(\text{roll an even number})=\dfrac{\color{red}3}{\color{blue}6}=\dfrac{\color{red}1}{\color{blue}2}

 

A LevelAQAEdexcelOCR

AND & OR

A\cap B means \mathbb(A) AND \mathbb(B) – that both event A and event B happen.

A\cup B means \mathbb(A) OR \mathbb(B) – that event A happens or event B happens (or both).

The probabilities of A\cap B and A\cup B are related by the formula:

\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)

 

These probabilities can be calculated with a Venn Diagram.

Example: considering dice again, if A is “rolls a multiple of 3” and B is “rolls 3 or 4”, then we have:

By looking at where the circles overlap, we notice there is one value, so A\cap B contains one value.

By looking at the circles in their entirety, we notice that there are 3 values contained within circle A or circle B, so A\cup B contains three values.

Since there are six values overall, we can conclude:

\mathbb{P}(A\cap B)=\dfrac{1}{6}

\mathbb{P}(A\cup B)=\dfrac{3}{6}=\dfrac{1}{2}

A LevelAQAEdexcelOCR

Complement of an Event

The complement of an event is the event that it doesn’t happen, for example if A is rolls a 3 or 4 then the complement of A, written A', is does not roll a 3 or 4, i.e. rolls a 1,2,5 or 6.

Since any event definitely either does or does not happen, the probability of an event and its complement must add to \mathbf{1}.

\mathbb{P}(A)+\mathbb{P}(A')=1

On a Venn Diagram, the complement of A is everything not contained within the circle representing A.

A LevelAQAEdexcelOCR

Two-Way Tables

Two way tables model two events, A and B, by displaying every combination of whether or not each one happens. They can either display frequency (which should be familiar from GCSE) or probability. If it displays probability, then the numbers in it excluding row totals and column totals must add to \mathbf{1}, and the final total in the bottom right must be 1. An example is given below.

Note: the totals are probabilities for one event, e.g. the total in the A column is the probability of event A. Using this, we can create the most general form of a two-way table:

A LevelAQAEdexcelOCR

Example Questions

There are 11 marbles, representing 11 possible outcomes, and 8 marbles that give the desired outcome, so the probability is \dfrac{8}{11}.

\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B)

 

\mathbb{P}(A)=0.4

 

\mathbb{P}(B)=0.3

 

\begin{aligned}\mathbb{P}(A\cup B)&=0.4+0.3-\mathbb{P}(A\cap B)\\[1.2em]&=0.7-\mathbb{P}(A\cap B)\end{aligned}

 

\mathbb{P}(A\cup B)+\mathbb{P}(A\cap B)=0.7

a)

 

b) 8 total events, 4 of them in circle A, so probability is \dfrac{4}{8}=\dfrac{1}{2}

 

c) 8 total events, 4 of them in circle B, so probability is \dfrac{4}{8}=\dfrac{1}{2}

 

d) This corresponds to where the circles overlap, which contains one event, so probability is \dfrac{1}{8}

 

e) This corresponds to both circles, which contain 7 events, so probability is \dfrac{7}{8}

 

f) 1 possibility does not lie within the Venn Diagram

\begin{aligned}\mathbb{P}(\text{complement})&=1-\mathbb{P}(\text{event})\\[1.2em]&=1-0.932\\[1.2em]&=0.068\end{aligned}

\mathbb{P}(A\cap B')=0.08

Additional Resources

MME

Exam Tips Cheat Sheet

A Level
MME

Formula Booklet

A Level

You May Also Like...

A Level Maths Revision Cards

The best A level maths revision cards for AQA, Edexcel, OCR, MEI and WJEC. Maths Made Easy is here to help you prepare effectively for your A Level maths exams.

£14.99
View Product

A Level Maths – Cards & Paper Bundle

A level maths revision cards and exam papers for Edexcel. Includes 2022 predicted papers based on the advance information released in February 2022! MME is here to help you study from home with our revision cards and practise papers.

From: £22.99
View Product

Transition Maths Cards

The transition maths cards are a perfect way to cover the higher level topics from GCSE whilst being introduced to new A level maths topics to help you prepare for year 12. Your ideal guide to getting started with A level maths!

£8.99
View Product