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Venn Diagram Probability
Here we will learn about Venn diagram probability, including how to calculate a probability using a Venn diagram.
There are also Venn diagram probability worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
What is Venn diagram probability?
Venn diagram probability is used to state the probability of or predict the possible outcomes of one or more event(s) occurring.
To do this we need to have a completed Venn diagram to be able to calculate probabilities from.
E.g.
Below is a Venn diagram describing the sets of odd numbers and prime numbers for the integer values in the universal set katex is not defined.
There are katex is not defined odd numbers. katex is not defined of these odd numbers are also prime numbers. The probability of selecting a number from the universal set that is odd, and prime, is katex is not defined out of the total number of values in the universal set, katex is not defined. The solution is therefore:
P(Odd and Prime) katex is not defined
Remember, because we are looking for a probability. This is usually expressed as a fraction or a decimal.
You must be aware of all of the set notation and symbols used for Venn diagrams.
What is a Venn diagram probability?
Conditional probability
When calculating probabilities from a Venn diagram, there may be conditions that need to be considered.
E.g. Let’s use the previous Venn diagram for the sets of odd numbers and prime numbers for the integer values in the universal set katex is not defined
If we wanted to calculate the probability of a prime number, given that the number is odd, we need to know the frequency of numbers that are prime and odd (katex is not defined), out of the total number of odd numbers (katex is not defined). The probability is therefore katex is not defined.
The condition we had was that the probability was out of another set, rather than the universal set (all of the values).
The questions in the exams will tend to use the word given to show that they are asking for a conditional probability.
In A level mathematics we use a the symbol | to represent the conditional probability in which we say given.
The formula for calculating a conditional probability is:
katex is not definedThe knowledge of this formula is not required for GCSE.
How to calculate probabilities from a Venn diagram
In order to calculate probabilities from a Venn diagram:
- Determine the parts of the Venn diagram that are in the subset.
- Calculate the frequency of the subset.
- Calculate the total frequency of the larger set.
- Write the probability as a fraction, and simplify.
Explain how to calculate probabilities from a Venn diagram
Venn diagram worksheet (includes Venn diagram probability)
Get your free venn diagram probability worksheet of 20+ Venn diagram questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREE Venn diagram worksheet (includes Venn diagram probability)
Get your free venn diagram probability worksheet of 20+ Venn diagram questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREERelated lessons on Venn diagram symbols
Venn diagram probability is part of our series of lessons to support revision on how to calculate probability. You may find it helpful to start with the main Venn diagram lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
Venn diagram probability examples
Example 1: two set Venn diagram
The universal set katex is not defined is an integerkatex is not defined. What is the probability of picking a number from the universal set that is a multiple of katex is not defined, but not a multiple of katex is not defined?
- Determine the parts of the Venn diagram that are in the subset.
The subset of the multiples of katex is not defined that are not a multiple of katex is not defined is the crescent of the left circle:
2Calculate the frequency of the subset.
The frequency of numbers within this subset is katex is not defined.
3Calculate the total frequency of the larger set.
The larger set is the universal set. The total frequency is therefore: katex is not defined
4Write the probability as a fraction, and simplify.
katex is not defined.
The probability of picking a multiple of katex is not defined that is not a multiple of katex is not defined from the universal set is katex is not defined.
Example 2: two set Venn diagram using symbols
The universal set katex is not defined {Quadrilaterals} . The Venn diagram shows the frequencies of quadrilaterals that belong to the two sets katex is not defined {rotational symmetry}, and katex is not defined {at least katex is not defined line of symmetry}. Calculate katex is not defined.
Determine the parts of the Venn diagram that are in the subset.
The subset of the union of katex is not defined and katex is not defined every item within both circles:
Calculate the frequency of the subset.
The frequency of numbers within this subset is katex is not defined
Calculate the total frequency of the larger set.
The larger set is the universal set. The total frequency is therefore: katex is not defined
Write the probability as a fraction, and simplify.
katex is not defined
katex is not defined
Example 3: two set Venn diagram using symbols
The universal set katex is not defined {Students in Year katex is not defined}. The Venn diagram shows the frequencies of students that belong to the two sets depending on which GCSE they study: katex is not defined{Geography}, and katex is not defined{History}. Calculate katex is not defined.
Determine the parts of the Venn diagram that are in the subset.
The complement to the intersection of katex is not defined and katex is not defined is the region:
Calculate the frequency of the subset.
The frequency of numbers within this subset is katex is not defined
Calculate the total frequency of the larger set.
The larger set is the universal set. The total frequency is therefore: katex is not defined
Write the probability as a fraction, and simplify.
katex is not defined
katex is not defined
Example 4: three set Venn diagram
katex is not defined people in a community group were asked about which of the three hobbies they enjoy. The three hobbies were Sewing, Pottery and Painting. The results are displayed in a three circle Venn diagram below.
What is the probability of picking a person at random that enjoys two or more hobbies?
Determine the parts of the Venn diagram that are in the subset.
As we are being asked about the frequency of two or more hobbies, we need to look at all of the values within the intersections of each circle.
Calculate the frequency of the subset.
Calculate the total frequency of the larger set.
Write the probability as a fraction, and simplify.
katex is not defined(two or more hobbies) katex is not defined
Example 5: three set Venn diagram using symbols
A clothes shop is carrying out some market research. They want to find out what accessories people wear when the weather turns cold. The results are shown below in a Venn diagram.
Calculate katex is not defined.
Determine the parts of the Venn diagram that are in the subset.
We want all the frequencies in the group katex is not defined, or those in the intersection of katex is not defined and katex is not defined:
Calculate the frequency of the subset.
Calculate the total frequency of the larger set.
Write the probability as a fraction, and simplify.
Example 6: conditional probability
katex is not defined students were asked if they had a computer or a games console. The results were recorded in the Venn diagram.
Calculate katex is not defined given that katex is not defined.
Determine the parts of the Venn diagram that are in the subset.
We need to identify the two subsets that are:
katex is not defined
katex is not defined
Calculate the frequency of the subset.
The frequency of values in the subset katex is not defined is katex is not defined
Calculate the total frequency of the larger set.
The frequency of values in the subset katex is not defined is katex is not defined
Write the probability as a fraction, and simplify.
katex is not defined given that katex is not defined
Common misconceptions
- Missing values
Make sure that each frequency required within the subset is included within the probability. This also includes values that are in the universal set and not in any other set.
- The probability is not a fraction
The probability is written as a total frequency instead of a fraction of the larger set.
- Using probability terminology incorrectly
The set of katex is not defined is written as katex is not defined which consists of a list of items or a frequency.
The probability of katex is not defined is written as katex is not defined and is a fraction. These two should not be confused.
- Conditional probability
The fraction is written out of the frequency of the universal set rather than the subset.
- Duplicated values
The items that are placed in the intersection are incorrectly added twice as it is assumed that the values are doubled as they belong to two sets. They only need to be counted once within a new subset.
Practice Venn diagram probability questions
1. The Venn diagram below shows the frequency of people who like certain genres of films. Using the Venn diagram below, calculate the probability of choosing someone at random who likes comedy films.
The frequency of people who like comedy films is katex is not defined
The total number of people asked is katex is not defined
The probability of choosing someone at random who likes comedy films is therefore katex is not defined
2. The Venn diagram below shows the frequency of people who like katex is not defined {ham} or katex is not defined {cheese} in a toasted sandwich.
Calculate katex is not defined
katex is not defined is the intersection of katex is not defined and katex is not defined (the overlapping section):
The frequency of people who like ham and cheese fillings is katex is not defined
The total number of people asked is katex is not defined
The probability of choosing someone at random who likes ham and cheese toasties is therefore katex is not defined
3. The Venn diagram below shows the frequency of people who have a katex is not defined {bath} or katex is not defined {shower} their main bathroom.
Calculate katex is not defined
katex is not defined is the union of items not in the set of katex is not defined or not in the set of katex is not defined
The frequency of people who have a bath or a shower but not both is katex is not defined
The total number of people asked is katex is not defined
The probability of choosing someone at random who likes ham and cheese toasties is therefore katex is not defined
4. A couple want to buy a horse. They would like to buy a horse that is katex is not defined {athletic}, katex is not defined {blue eyes}, katex is not defined {piebald cob}. They record the frequencies of their search into a Venn diagram.
Calculate the probability of them buying a horse that has only one of the three requirements.
The subsets within the Venn diagram that contain only one of the three requirements are:
The frequency of horses in the subset is katex is not defined
The total number of horses is katex is not defined
The probability of buying a horse that has at least one of the three requirements is therefore katex is not defined
5. A vexillologist collects flags. He wants to analyse the contents of the flags in his collection into the following three sets:
- katex is not defined{Stars}
- katex is not defined{Stripes}
- katex is not defined{An Image}
He writes up his findings in a Venn diagram.
Calculate katex is not defined
The frequencies in each subset are:
katex is not defined
katex is not defined
We therefore have:
katex is not defined
6. A cafe serves breakfast butties. They count the frequency of butties that were made throughout the week with each ingredient:
- katex is not defined{Bacon}
- katex is not defined{Egg}
- katex is not defined{Sausage}
The results are written into a Venn diagram.
Calculate the probability the butties contain bacon and eggs given that they contain eggs.
The frequencies in each subset are:
katex is not defined
katex is not defined
We therefore have:
katex is not defined
Venn diagram probability GCSE questions
1. A reptile house records the toxicity of their katex is not defined different species of wildlife into the Venn diagram below. The two sets represent whether the animal is katex is not defined {venomous} or katex is not defined {poisonous}.
The Asian Tiger snake is the only animal that is both venomous and poisonous.
(a) Complete the Venn diagram.
(b) An animal breaks out of a vivarium. What is the probability that this species is venomous?
(3 marks)
(a)
katex is not defined written in the intersection
(1)
katex is not defined written in the universal set
(1)
(b)
katex is not defined
(1)
2. (a) A car salesman is researching the number of cars in the area that are either katex is not defined {Electric} or katex is not defined {Petrol}. He displays the results into the Venn diagram.
How many cars were recorded in the Venn diagram?
(b) Calculate the probability of katex is not defined given katex is not defined Write your answer as a fraction in the simplest form.
(5 marks)
(a)
katex is not defined
(1)
(b)
katex is not defined
(1)
katex is not defined(1)
katex is not defined(1)
katex is not defined(1)
3. katex is not defined people participated in a local marathon. The Venn diagram below splits all of the participants into the three sets: katex is not defined {Completed}, katex is not defined {Fancy Dress} and katex is not defined {Ran all of the way}.
(a) Given that katex is not defined of participants wore fancy dress, and katex is not defined of all runners completed the marathon, complete the Venn diagram.
(b) Calculate katex is not defined correct to katex is not defined decimal places.
(7 marks)
(a)
katex is not defined written in the intersection of Completed and wore Fancy dress
(1)
katex is not defined written in the intersection of Fancy dress and Ran
(1)
katex is not defined written in the universal set outside of set katex is not defined and katex is not defined
(1)
(b)
katex is not defined
(1)
katex is not defined
(1)
katex is not defined
(1)
katex is not defined
(1)
Learning checklist
You have now learned how to:
- Calculate and interpret conditional probabilities through representation using expected frequencies with Venn diagrams
The next lessons are
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