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Lesson 10 of 13
  • Year 9

Calculating theoretical probabilities from Venn diagrams (two events)

I can calculate and use theoretical probabilities for combined events using Venn diagrams (2 events).

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Lesson 10 of 13
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New
  • Year 9

Calculating theoretical probabilities from Venn diagrams (two events)

I can calculate and use theoretical probabilities for combined events using Venn diagrams (2 events).

Download all resources
Share activities with pupils

These resources will be removed by end of Summer Term 2025.

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These resources were created for remote use during the pandemic and are not designed for classroom teaching.

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Lesson details

Key learning points

  1. The probability of an outcome can be found by considering a Venn diagram showing all possible outcomes for two events.
  2. The probability of a set of outcomes can be found by using a Venn diagram showing all possible outcomes for two events.
  3. The probability of a set of outcomes can be found using a Venn diagram, even when the outcomes are not equally likely.

Keywords

  • Theoretical probability - A theoretical probability is a probability based on counting the number of desired outcomes from a sample space where all individual outcomes are equally likely.

  • Venn diagram - Venn diagrams are a representation used to model statistical/probability questions. Commonly circles are used to represent events.

Common misconception

Pupils may struggle with finding the probability of A or B, and may count the outcomes that belong to A and B twice.

If you use an example, such as visiting particular countries, ask the pupils how someone would respond to the question 'have you visited X or Y' if they have in fact visited both countries.


To help you plan your year 9 maths lesson on: Calculating theoretical probabilities from Venn diagrams (two events), download all teaching resources for free and adapt to suit your pupils' needs...

If the pupils have copies of the Venn diagrams or a mini-whiteboard they can go through each outcome to see if they are to be included in the combined event or not. Over time the pupils should notice the regions that are being included in particular questions.
Teacher tip

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Prior knowledge starter quiz

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6 Questions

Q1.
There are two trials. In Trial 1, a fair spinner with {A, B} is spun twice. In Trial 2, a fair spinner with {B, C} is spun twice. Which statement is true?

The most likely outcome is {B,B}.
Correct answer: B is the most likely letter to appear in the trials.
C is the least likely letter to appear in the trials.

Q2.
If the probability of an event (A) is $$/frac{2}{5}$$, and the experiment is repeated 25 times, what is the theoretical number of times event (A) will occur?

5
Correct answer: 10
15
20
25

Q3.
A random character is selected from the word MISSISSIPPI, which of the following statements are true?

The letter selected would most likely be a vowel
Correct answer: The letter selected would most likely be a consonant
The letter selected would most likely be an 'S'
Correct answer: The letter selected would least likely be an 'M'

Q4.
If the probability of an event (A) is $$/frac{1}{8}$$, and the experiment is repeated 40 times, what is the theoretical number of times event (A) will occur?

1
8
Correct answer: 5
18

Q5.
If you roll a fair six sided die twice and sum the results, what is the most likely sum below?

Correct answer: 7
6
8

Q6.
If you roll a fair six sided die twice and sum the results, what is the most likely sum below?

an even sum
an odd sum
Correct answer: An even or an odd sum are equally likely

Assessment exit quiz

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6 Questions

Q1.
Using this Venn diagram, what is the probability of an object being randomly selected from B?

An image in a quiz
Correct answer: $$\frac{5}{7}$$
$$\frac{2}{7}$$
$$\frac{2}{5}$$
$$\frac{5}{6}$$

Q2.
Using this Venn diagram, what is the probability of an object being randomly selected that is in A and B?

An image in a quiz
$$\frac{5}{7}$$
Correct answer: $$\frac{3}{7}$$
$$\frac{1}{2}$$
$$\frac{6}{7}$$

Q3.
Using the diagram, what is P(A and not B)?

An image in a quiz
$$\frac{5}{7}$$
$$\frac{3}{7}$$
Correct answer: $$\frac{1}{7}$$
$$\frac{4}{7}$$

Q4.
Using the diagram, what is P(not A and not B)?

An image in a quiz
$$\frac{5}{7}$$
$$\frac{3}{7}$$
$$\frac{4}{7}$$
Correct answer: $$\frac{1}{7}$$

Q5.
Using the diagram, what is P(not A and not B)?

An image in a quiz
Correct answer: $$\frac{2}{7}$$
$$\frac{3}{7}$$
$$\frac{4}{7}$$
$$\frac{1}{7}$$

Q6.
Using the diagram, what is P(A and B)?

An image in a quiz
$$\frac{2}{7}$$
$$\frac{3}{7}$$
$$\frac{4}{7}$$
$$\frac{1}{7}$$