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Maths · Probability

Venn diagrams and set notation

Use set notation, draw and complete Venn diagrams, and use them to find probabilities, including conditional probabilities at Higher tier.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    List the even numbers from 1 to 10.

    Show answerHide answer

    2, 4, 6, 8, 10

  2. 2

    List the multiples of 3 from 1 to 10.

    Show answerHide answer

    3, 6, 9

  3. 3

    What number is in both lists above?

    Show answerHide answer

    6

  4. 4

    Work out \(40 - 29\).

    Show answerHide answer

    11

  5. 5

    Simplify \(\dfrac{8}{16}\).

    Show answerHide answer

    \(\dfrac{1}{2}\)

Learning Objectives

  1. 1Use set notation: \(\in\), \(\cup\), \(\cap\), \(A'\), \(\xi\) and \(n(A)\).
  2. 2Draw and complete a Venn diagram.
  3. 3Shade the region for a set expression.
  4. 4Find probabilities from a Venn diagram, including conditional ones (Higher).

Set Notation

Learn these symbols.

  • \(\xi\)

    Meaning: The universal set: everything being considered. Example: \(\xi = \{1, 2, ..., 10\}\)

  • \(A \cup B\)

    Meaning: Union: in A or B or both. Example: \(\{2, 3, 4, 6, 8, 9, 10\}\)

  • \(A \cap B\)

    Meaning: Intersection: in both A and B. Example: \(\{6\}\)

  • \(A'\)

    Meaning: Complement: not in A. Example: \(\{1, 3, 5, 7, 9\}\)

  • \(n(A)\)

    Meaning: The number of elements in A. Example: \(n(A) = 5\)

  • \(\in\)

    Meaning: "is an element of". Example: \(4 \in A\)

Sets of Numbers

\(\xi = \{1, 2, 3, ..., 10\}\), \(A = \) the even numbers and \(B = \) the multiples of 3. List \(A\), \(B\), \(A \cap B\), \(A \cup B\) and \(A'\).

Show the solutionHide the solution
  1. 1 \(A\) \(\{2, 4, 6, 8, 10\}\)
  2. 2 \(B\) \(\{3, 6, 9\}\)
  3. 3 In both \(A \cap B = \{6\}\)
  4. 4 In either \(A \cup B = \{2, 3, 4, 6, 8, 9, 10\}\)
  5. 5 Not in A \(A' = \{1, 3, 5, 7, 9\}\)

AnswerSee the lists above; \(n(A \cup B) = 7\).

Completing a Venn Diagram

Start in the middle and work outwards.

  1. 1 Fill in the overlap first

    The number in both sets
  2. 2 Work out the rest of each set

    Subtract the overlap from each total
  3. 3 Fill the outside

    Universal total minus everything inside the circles
  4. 4 Check

    All the numbers add up to the universal total

A Survey

40 students were asked about music (M) and art (R). 22 take music, 15 take art and 8 take both. Complete the Venn diagram and find the number who take neither.

Show the solutionHide the solution
  1. 1 Both 8
  2. 2 Music only \(22 - 8 = 14\)
  3. 3 Art only \(15 - 8 = 7\)
  4. 4 Inside the circles \(14 + 8 + 7 = 29\)
  5. 5 Neither \(40 - 29 = 11\)

AnswerMusic only 14, both 8, art only 7, neither 11.

Probabilities from a Venn Diagram

For the survey, a student is chosen at random. Find (a) \(P(\text{neither})\), (b) \(P(M \cup R)\).

Show the solutionHide the solution
  1. 1 (a) Neither \(\dfrac{11}{40}\)
  2. 2 (b) Music or art (or both) \(14 + 8 + 7 = 29\), so \(\dfrac{29}{40}\)

Answer(a) \(\dfrac{11}{40}\) (b) \(\dfrac{29}{40}\)

Given That

For the survey, find \(P(M \mid R)\): the probability a student takes music, given that they take art.

Show the solutionHide the solution
  1. 1 "Given that they take art" restricts to the art circle 15 students
  2. 2 Of these, the number who also take music 8
  3. 3 Probability \(\dfrac{8}{15}\)

Answer\(\dfrac{8}{15}\)

Class Survey

In a class of 30 students, 18 have a dog, 12 have a cat and 5 have both. Draw a Venn diagram, and find the probability that a randomly chosen student has (a) a dog only (b) neither (c) a cat given that they have a dog.

1. Fill in the overlap first.

2. Subtract for the other regions.

3. Use the right total.

A good answer shows: Dog only \(18 - 5 = 13\), cat only \(12 - 5 = 7\), both 5, neither \(30 - 25 = 5\). (a) \(\dfrac{13}{30}\) (b) \(\dfrac{5}{30} = \dfrac{1}{6}\) (c) \(\dfrac{5}{18}\).

Can I...?

  1. 1Use the set notation symbols.
  2. 2List the elements of \(A \cap B\) and \(A \cup B\).
  3. 3Find \(A'\).
  4. 4Fill in a Venn diagram from totals.
  5. 5Shade the correct region.
  6. 6Find a probability from a Venn diagram.
  7. 7Find \(P(A \mid B)\) (Higher).
  8. 8Check the total.

Summary & Exam Focus

  • \(A \cup B\): in A or B. \(A \cap B\): in both. \(A'\): not in A.
  • Fill the overlap first, then each circle, then the outside.
  • The numbers must add up to the universal total.
  • Higher: \(P(A \mid B)\) uses B as the new total.

Exam focus

40 students were asked about music and art. 22 take music, 15 take art and 8 take both. Draw a Venn diagram to show this information. (4 marks) (4 marks)

Start with the overlap (8), then subtract it from each circle total. Finish with the number outside.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Set
A collection of items.
Element
One item in a set.
Universal set
The set of everything being considered, \(\xi\).
Union
Items in one set or the other or both, \(\cup\).
Intersection
Items in both sets, \(\cap\).
Complement
Items not in the set, \(A'\).

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    \(\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\), \(A = \{\text{even numbers}\}\) and \(B = \{\text{multiples of } 3\}\). List the members of \(A \cap B\).

    Show answerHide answer

    Model answer

    \(A \cap B = \{6\}\).

    Mark scheme

    • \(A = \{2, 4, 6, 8, 10\}\) and \(B = \{3, 6, 9\}\) — M1
    • \(\{6\}\) — A1
  2. Question 2 Non-calculator 2 marks

    Using the same sets, list the members of \(A'\).

    Show answerHide answer

    Model answer

    \(A' = \{1, 3, 5, 7, 9\}\).

    Mark scheme

    • Odd numbers seen — M1
    • \(\{1, 3, 5, 7, 9\}\) — A1
  3. Question 3 Non-calculator 4 marks

    40 students were asked whether they take music (M) and art (R). 22 take music, 15 take art and 8 take both. Complete the Venn diagram.

    A Venn diagram with sets M and R inside a universal set of 40 students, with 8 written in the overlap and the other regions blank.
    Show answerHide answer

    Model answer

    M only 14, overlap 8, R only 7, outside the circles 11.

    Mark scheme

    • Overlap 8 — B1
    • M only 14 — B1
    • R only 7 — B1
    • Outside 11 — B1
  4. Question 4 Non-calculator 2 marks

    One of the 40 students is chosen at random. Use your Venn diagram to work out the probability that the student takes neither music nor art.

    Show answerHide answer

    Model answer

    \(\dfrac{11}{40}\).

    Mark scheme

    • 11 — M1
    • \(\dfrac{11}{40}\) — A1
  5. Question 5 Non-calculator 2 marks

    Write down the number of students in \(M \cup R\).

    Show answerHide answer

    Model answer

    \(14 + 8 + 7 = 29\).

    Mark scheme

    • Adding the three regions — M1
    • 29 — A1
  6. Question 6 Non-calculator 3 marks

    One of the 40 students is chosen at random. Given that the student takes art, work out the probability that they also take music.

    Show answerHide answer

    Model answer

    15 students take art, and 8 of these also take music. The probability is \(\dfrac{8}{15}\).

    Mark scheme

    • 15 as the total — M1
    • \(\dfrac{8}{15}\) — A1
    • A correct conclusion — C1

Quick check

  1. What does \(A \cap B\) mean?

    1. AIn A or B
    2. BIn A but not B
    3. CIn both A and B
    4. DNot in A
    Show answerHide answer

    C: In both A and B

    The intersection: in both A and B.

  2. What does \(A'\) mean?

    1. AIn A
    2. BNot in A
    3. CIn A and B
    4. DTwice A
    Show answerHide answer

    B: Not in A

    The complement: everything in \(\xi\) that is not in A.

  3. \(\xi = \{1, 2, 3, 4, 5\}\) and \(A = \{1, 2\}\). What is \(A'\)?

    1. A\(\{1, 2\}\)
    2. B\(\{1, 2, 3\}\)
    3. C\(\{4, 5\}\)
    4. D\(\{3, 4, 5\}\)
    Show answerHide answer

    D: \(\{3, 4, 5\}\)

    The elements of \(\xi\) not in A: \(\{3, 4, 5\}\).

  4. In a Venn diagram with total 30, 12 are in A only, 5 in both and 7 in B only. How many are in neither?

    1. A6
    2. B24
    3. C5
    4. D12
    Show answerHide answer

    A: 6

    \(30 - (12 + 5 + 7) = 6\).

  5. What does \(n(A)\) stand for?

    1. AThe name of A
    2. BThe number of elements in A
    3. CThe complement of A
    4. DThe union of A
    Show answerHide answer

    B: The number of elements in A

    The number of elements in set A.

  6. \(P(A \mid B)\) is read as...

    1. AThe probability of A and B
    2. BThe probability of A or B
    3. CThe probability of A given B
    4. DThe probability of not A
    Show answerHide answer

    C: The probability of A given B

    The probability of A given that B has happened.

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