[discia] GCSE Maths › Probability › Venn Diagrams

Venn Diagrams

discia.org 10 questions · 29 marks
Name Date
  1. Either paper 2 marks

    \(A\) and \(B\) are two events.

    (a)

    What does \(A \cup B\) represent in \(P(A \cup B)\)?

    Circle your answer.

    - not A or not B, or both - A but not B - A and B - A or B or both

    (b)

    Circle the expression that means the probability of A and not B.

    - \(P(A \cap B')\) - \(P(A' \cup B)\) - \(P(A' \cap B)\) - \(P(A \cup B')\)

    Show answer
    (a) A or B or both
    (b) \(P(A \cap B')\)
    Another like this
  2. Either paper 4 marks

    In a group of 35 children, 23 have a dog and 17 have a cat.

    2 of the children have neither.

    You may use the Venn diagram to help you.

    DogCatξ
    (a)

    How many of the children have both a dog and a cat?

    (b)

    One of the children is chosen at random.

    Find the probability that this person is one of those who have a dog but not a cat.

    Show answer
    (a) \(7\)
    (b) \(\frac{16}{35}\)
    Another like this
  3. Either paper 5 marks

    The Venn diagram shows information about a group of members.

    GymSwimming1335ξ3
    (a)

    How many members are in the group altogether?

    (b)

    One of the members is chosen at random.

    Calculate the probability that this person is one of those who go to the gym.

    (c)

    One of the members is chosen at random.

    Calculate the probability that this person is in exactly one of the two circles.

    Show answer
    (a) \(24\)
    (b) \(\frac{2}{3}\)
    (c) \(\frac{3}{4}\)
    Another like this
  4. Either paper 1 mark
    The Venn diagram shows the sets \(C\) and \(D\). CDξ Which of these represents the shaded region? Circle your answer.
    \(C\)\(C' \cap D\)\(C \cap D\)\(C \cup D'\)
    Show answer
    \(C \cap D\)
    Another like this
  5. Either paper 6 marks

    \(\xi\) is the set of whole numbers from 1 to 15.

    \(A = \{ 1,\ 2,\ 4,\ 5,\ 10 \}\) and \(B = \{ 2,\ 3,\ 5,\ 7,\ 11,\ 13 \}\) are two of its subsets.

    Every number in \(\xi\) goes in exactly one region of the Venn diagram.

    ABξ
    (a)

    Write down the numbers that are in both \(A\) and \(B\).

    (b)

    Write down the numbers that are in \(A\) but not in \(B\).

    (c)

    Write down the numbers that are in \(B\) but not in \(A\).

    (d)

    Write down the numbers that are in neither \(A\) nor \(B\).

    (e)

    A number is chosen at random from \(\xi\).

    Find \(P(A \cap B)\).

    Show answer
    (a) \(2,\ 5\)
    (b) \(1,\ 4,\ 10\)
    (c) \(3,\ 7,\ 11,\ 13\)
    (d) \(6,\ 8,\ 9,\ 12,\ 14,\ 15\)
    (e) \(\frac{2}{15}\)
    Another like this
  6. Either paper 1 mark

    The shaded region of the Venn diagram shows a set.

    ABξ

    Use set notation to write down the shaded set.

    Show answer
    \(A \cup B'\)
    Another like this
  7. Either paper 1 mark

    Shade the region \(A \cap B'\) on the Venn diagram.

    ABξ
    Show answer
    Shade the part of A that is outside B.
    ABξ
    Another like this
  8. Either paper 5 marks

    55 members were asked which of the gym, the pool and the studio they use.

    The Venn diagram shows the results.

    GymPoolStudio741064811ξ5
    (a)

    How many of the members use all three?

    (b)

    How many of the members use exactly one of the three?

    (c)

    One of the members is chosen at random.

    Find the probability that they use the gym and the studio.

    Show answer
    (a) \(4\)
    (b) \(28\)
    (c) \(\frac{2}{11}\)
    Another like this
  9. Either paper 2 marks

    In a group of 88 students, 26 study both Art and French.

    41 study Art.

    44 study French.

    Grace draws this Venn diagram to represent the information.

    ξ = the group of 88 students

    ArtFrench152644ξ

    Make two criticisms of her diagram. For each criticism, say which region is wrong and what number it should contain.

    Criticism 1

    Criticism 2

    Show answer
    Criticism 1: the French-only region should contain 18: the total for French was written as the French-only number, so the overlap is counted twice and the number who study both must be taken off. Criticism 2: the outside region should contain 29: it is blank, and it should hold the people who study neither subject, which is the group total minus everyone in the two circles.
    Another like this
  10. Either paper 2 marks

    In a group of 33 students, 11 play hockey, 18 play tennis and 10 do not play hockey or tennis.

    Owen draws this Venn diagram to represent the information.

    ξ = the group of 33 students

    HockeyTennis1118ξ10

    Make two criticisms of his diagram. For each criticism, say which region is wrong and what number it should contain.

    Criticism 1

    Criticism 2

    Show answer
    Any two of: the overlap region is empty but should contain 6, because the numbers 11, 18 and 10 add up to 39, not 33; the Hockey-only region should contain 5, not 11, because 11 includes the overlap; the Tennis-only region should contain 12, not 18, because 18 includes the overlap.
    Another like this
discia.org Venn Diagrams · sheet 1
Answers Venn Diagrams · sheet 1
  1. (a) A or B or both
    (b) \(P(A \cap B')\)
  2. (a) \(7\)
    (b) \(\frac{16}{35}\)
  3. (a) \(24\)
    (b) \(\frac{2}{3}\)
    (c) \(\frac{3}{4}\)
  4. \(C \cap D\)
  5. (a) \(2,\ 5\)
    (b) \(1,\ 4,\ 10\)
    (c) \(3,\ 7,\ 11,\ 13\)
    (d) \(6,\ 8,\ 9,\ 12,\ 14,\ 15\)
    (e) \(\frac{2}{15}\)
  6. \(A \cup B'\)
  7. Shade the part of A that is outside B.
    ABξ
  8. (a) \(4\)
    (b) \(28\)
    (c) \(\frac{2}{11}\)
  9. Criticism 1: the French-only region should contain 18: the total for French was written as the French-only number, so the overlap is counted twice and the number who study both must be taken off. Criticism 2: the outside region should contain 29: it is blank, and it should hold the people who study neither subject, which is the group total minus everyone in the two circles.
  10. Any two of: the overlap region is empty but should contain 6, because the numbers 11, 18 and 10 add up to 39, not 33; the Hockey-only region should contain 5, not 11, because 11 includes the overlap; the Tennis-only region should contain 12, not 18, because 18 includes the overlap.
discia.org Venn Diagrams · sheet 1