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Exam questions · Maths · Probability

Venn Diagrams and Set Notation

  • 6 exam questions
  • 19 marks
  • 9 quick checks
  1. 1 Write down [2 marks]

    \(\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\). \(A\) is the set of even numbers and \(B = \{2, 3, 5, 7\}\). (a) Write down \(A \cap B\). (1 mark) (b) Write down \(A \cup B\). (1 mark)

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    Model answer

    (a) \(A \cap B = \{2\}\). (b) \(A \cup B = \{2, 3, 4, 5, 6, 7, 8, 10\}\).

    Mark scheme

    • (a) \(\{2\}\) — B1
    • (b) \(\{2, 3, 4, 5, 6, 7, 8, 10\}\) — B1
  2. 2 Work out [4 marks]

    There are 40 students in a class. 22 play football and 19 play tennis. The incomplete Venn diagram shows 6 who play both sports and 5 who play neither. (a) Complete the Venn diagram. (2 marks) (b) A student is chosen at random. Work out the probability that the student plays exactly one of the two sports. (2 marks)

    A Venn diagram of football and tennis with 6 in the overlap, 5 outside both circles and the other two regions empty.
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    Model answer

    (a) Football only is \(22 - 6 = 16\) and tennis only is \(19 - 6 = 13\). Check: \(16 + 6 + 13 + 5 = 40\). (b) \(\dfrac{16 + 13}{40} = \dfrac{29}{40}\).

    Mark scheme

    • (a) 16 — B1
    • (a) 13 — B1
    • (b) \(16 + 13 = 29\) seen — M1
    • (b) \(\dfrac{29}{40}\) — A1
  3. 3 Work out [3 marks]

    \(\xi = \{1, 2, 3, \ldots, 12\}\). \(A\) is the set of multiples of 3 and \(B\) is the set of factors of 12. (a) List the members of \(A \cap B\). (1 mark) (b) Work out \(n((A \cup B)')\). (2 marks)

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    Model answer

    (a) \(A = \{3, 6, 9, 12\}\) and \(B = \{1, 2, 3, 4, 6, 12\}\), so \(A \cap B = \{3, 6, 12\}\). (b) \(A \cup B = \{1, 2, 3, 4, 6, 9, 12\}\), which has 7 members, so \(n((A \cup B)') = 12 - 7 = 5\).

    Mark scheme

    • (a) \(\{3, 6, 12\}\) — B1
    • (b) Lists or counts 7 members of \(A \cup B\) — M1
    • (b) 5 — A1
  4. 4 Work out [3 marks]

    In a group of 50 people, 30 like tea and 25 like coffee. 8 people like neither drink. Work out the number of people who like both tea and coffee.

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    Model answer

    The number who like at least one drink is \(50 - 8 = 42\). Then \(30 + 25 - 42 = 13\) like both.

    Mark scheme

    • \(50 - 8 = 42\) — M1
    • \(30 + 25 - 42\) — M1
    • 13 — A1
  5. 5 Work out [4 marks]

    In a Venn diagram of 52 people, the number in set \(A\) only is \(2x\), the number in both sets is \(x\), the number in set \(B\) only is \(x + 4\), and 8 people are in neither set. (a) Work out the value of \(x\). (2 marks) (b) A person is chosen at random. Work out the probability that the person is in set \(A\). Give your answer in its simplest form. (2 marks)

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    Model answer

    (a) \(2x + x + x + 4 + 8 = 52\), so \(4x + 12 = 52\) and \(x = 10\). (b) Set \(A\) has \(2x + x = 30\) people, so the probability is \(\dfrac{30}{52} = \dfrac{15}{26}\).

    Mark scheme

    • (a) \(4x + 12 = 52\) — M1
    • (a) \(x = 10\) — A1
    • (b) \(\dfrac{30}{52}\) — M1
    • (b) \(\dfrac{15}{26}\) — A1
  6. 6 Work out [3 marks]

    There are 30 students in a class. 18 study French and 12 study Spanish. 5 study both. A student is chosen at random. Work out the probability that the student studies neither French nor Spanish.

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    Model answer

    At least one language: \(18 + 12 - 5 = 25\). Neither: \(30 - 25 = 5\). The probability is \(\dfrac{5}{30} = \dfrac{1}{6}\).

    Mark scheme

    • \(18 + 12 - 5 = 25\) — M1
    • 5 students study neither — A1
    • \(\dfrac{1}{6}\) — A1

Quick check

  1. 1

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

    1. AThe items in \(A\) or \(B\)
    2. BThe items not in \(A\)
    3. CThe items only in \(A\)
    4. DThe items in both \(A\) and \(B\)
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    D: The items in both \(A\) and \(B\)

    \(\cap\) is the intersection, the overlap.

  2. 2

    What does \(A \cup B\) mean?

    1. AThe items in both \(A\) and \(B\)
    2. BThe items not in \(B\)
    3. CThe items in \(A\) or \(B\) or both
    4. DThe items only in \(B\)
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    C: The items in \(A\) or \(B\) or both

    \(\cup\) is the union, everything in either circle.

  3. 3

    What does \(A'\) mean?

    1. AThe items in \(A\) only
    2. BThe items not in \(A\)
    3. CThe items in both sets
    4. DThe number of items in \(A\)
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    B: The items not in \(A\)

    \(A'\) is the complement of \(A\).

  4. 4

    \(A = \{2, 4, 6, 8, 10\}\) and \(B = \{3, 6, 9\}\). What is \(A \cap B\)?

    1. A\(\{6\}\)
    2. B\(\{3, 6, 9\}\)
    3. C\(\{2, 3, 4, 6, 8, 9, 10\}\)
    4. D\(\{\}\)
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    A: \(\{6\}\)

    Only 6 is in both sets.

  5. 5

    18 students play football and 6 of them also play tennis. How many play football only?

    1. A\(18\)
    2. B\(24\)
    3. C\(6\)
    4. D\(12\)
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    D: \(12\)

    \(18 - 6 = 12\).

  6. 6

    30 students: 18 play football, 14 play tennis and 6 play both. How many play neither?

    1. A\(2\)
    2. B\(8\)
    3. C\(4\)
    4. D\(10\)
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    C: \(4\)

    \(12 + 6 + 8 = 26\) play at least one, so \(30 - 26 = 4\).

  7. 7

    In the same survey, what is the probability that a student chosen at random plays football only?

    1. A\(\dfrac{3}{5}\)
    2. B\(\dfrac{2}{5}\)
    3. C\(\dfrac{1}{5}\)
    4. D\(\dfrac{18}{30}\)
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    B: \(\dfrac{2}{5}\)

    \(\dfrac{12}{30} = \dfrac{2}{5}\). The fraction \(\dfrac{18}{30}\) would include those who play both.

  8. 8

    In a class of 40, 22 play football, 19 play tennis and 5 play neither. How many play both?

    1. A\(6\)
    2. B\(3\)
    3. C\(9\)
    4. D\(17\)
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    A: \(6\)

    \(40 - 5 = 35\) play at least one, and \(22 + 19 - 35 = 6\).

  9. 9

    \(n(A) = 15\), \(n(B) = 12\) and \(n(A \cap B) = 5\). What is \(n(A \cup B)\)?

    1. A\(27\)
    2. B\(32\)
    3. C\(17\)
    4. D\(22\)
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    D: \(22\)

    \(15 + 12 - 5 = 22\).