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13.4 Venn Diagrams

GCSE Maths Foundation · Edexcel · Chapter 13: Probability

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: sets, set notation, drawing Venn diagrams and finding probabilities from them. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Understand sets and the symbols ξ, ∈, { }, ∩, ∪ and A′
  • Sort the members of two sets into a Venn diagram
  • Complete a Venn diagram showing frequencies
  • Work out probabilities from a Venn diagram

Key words

  • Set: A collection of things, called members or elements, written in curly brackets: A = {2, 4, 6}.
  • Universal set, ξ: Everything being considered. It is the rectangle in a Venn diagram.
  • Intersection, A ∩ B: Members in A and in B (the overlap).
  • Union, A ∪ B: Members in A or B or both (everything inside the circles).
  • Complement, A′: Members not in A (everything outside circle A).

Sets and set notation

A set can be listed or described. The symbol ∈ means “is a member of”, and n(A) means the number of members of A.

Worked example: writing sets

ξ = {whole numbers from 1 to 12}. A = {even numbers}. B = {multiples of 3}.
A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12}.
9 ∈ B, but 9 is not in A. n(A) = 6 and n(B) = 4.

Answer: A = {2, 4, 6, 8, 10, 12}, B = {3, 6, 9, 12}

Drawing a Venn diagram

Work in this order: (1) the overlap – numbers in both sets; (2) the rest of each circle; (3) everything else in ξ goes outside the circles. Every member of ξ appears exactly once.

Worked example: reading the diagram

A ∩ B = {6, 12} (in both circles).
A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12} (anywhere in the circles).
B′ = {1, 2, 4, 5, 7, 8, 10, 11} (everything outside B).

Answer: A ∩ B = {6, 12}

Shading regions

It helps to picture each piece of notation as a shaded region.

Key fact: ∩ looks like an “n” for and: the overlap only. ∪ looks like a “u” for union: everything in either circle.

Venn diagrams with frequencies

Often a Venn diagram shows how many things are in each region instead of listing them. Again, start with the overlap and work outwards.

Worked example: guitar and piano

In a group of 40 students, 23 play the guitar (G), 15 play the piano (P) and 6 play both.
Guitar only = 23 − 6 = 17. Piano only = 15 − 6 = 9.
Neither = 40 − (17 + 6 + 9) = 40 − 32 = 8.

Answer: see the diagram

Probability from a Venn diagram

P(event) = number in the region ÷ total number in ξ.

Worked example: using the music diagram

One of the 40 students is chosen at random.
P(plays both) = 6/40 = 3/20.
P(plays exactly one instrument) = (17 + 9)/40 = 26/40 = 13/20.
P(does not play the piano) = (17 + 8)/40 = 25/40 = 5/8.

Answer: 3/20, 13/20, 5/8

Common mistakes

  • Putting the overlap numbers in each circle as well as the middle, so they are counted twice.
  • Forgetting the region outside the circles. Members of ξ that are in neither set still go in the rectangle.
  • Writing 23 in the “guitar only” part when 23 is the total for the whole circle. Subtract the overlap first.
  • Mixing up ∩ (and) with ∪ (or).

Revision questions

Try these, then check your answers below each one.

Question 1. A Venn diagram shows pets owned by 30 people. Dog only: 12. Both a dog and a cat: 5. Cat only: 9. Neither: 4. How many people own a cat?

Answer: 14

Working: 9 + 5 = 14.

Question 2. Use the same Venn diagram (dog only 12, both 5, cat only 9, neither 4). A person is chosen at random. Work out P(owns neither). Give a fraction in its simplest form.

Answer: 2/15

Working: 4/30 = 2/15.

Question 3. What does A' mean? (Choose from: Everything in A, Everything not in A, Everything in A and B, Everything in A or B.)

Answer: Everything not in A

Working: A' is the complement of A: everything in ξ that is not in A.

Related lessons

Continue in the full course

This lesson is section 13.4 (EGCMF1304) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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