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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]

    \(\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}\). \(A\) is the set of odd numbers and \(B\) is the set of square numbers. (a) Write down \(A \cap B\). [1 mark] (b) Work out \(n(A \cup B)\). [1 mark]

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    Model answer

    (a) \(A = \{1, 3, 5, 7, 9\}\) and \(B = \{1, 4, 9\}\), so \(A \cap B = \{1, 9\}\). (b) \(A \cup B = \{1, 3, 4, 5, 7, 9\}\), so \(n(A \cup B) = 6\).

    Mark scheme

    • (a) \(\{1, 9\}\) — B1
    • (b) 6 — B1
  2. 2 Work out [4 marks]

    44 pupils each choose art, music, both or neither. 25 choose art and 20 choose music. The incomplete Venn diagram shows 9 pupils in both and 8 in neither. (a) Complete the Venn diagram. [2 marks] (b) A pupil is chosen at random. Work out the probability that the pupil chose music but not art. [2 marks]

    A Venn diagram of art and music with 9 in the overlap, 8 outside both circles and the other two regions empty.
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    Model answer

    (a) Art only is \(25 - 9 = 16\) and music only is \(20 - 9 = 11\). Check: \(16 + 9 + 11 + 8 = 44\). (b) \(\dfrac{11}{44} = \dfrac{1}{4}\).

    Mark scheme

    • (a) 16 — B1
    • (a) 11 — B1
    • (b) \(\dfrac{11}{44}\) — M1
    • (b) \(\dfrac{1}{4}\) — A1
  3. 3 Work out [3 marks]

    \(\xi = \{1, 2, 3, \ldots, 15\}\). \(A\) is the set of multiples of 2 and \(B\) is the set of multiples of 5. (a) Write down \(A \cap B\). [1 mark] (b) A number is chosen at random from \(\xi\). Work out the probability that it is in \(A \cup B\). [2 marks]

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    Model answer

    (a) \(A \cap B = \{10\}\). (b) \(A \cup B = \{2, 4, 5, 6, 8, 10, 12, 14, 15\}\), which has 9 members, so the probability is \(\dfrac{9}{15} = \dfrac{3}{5}\).

    Mark scheme

    • (a) \(\{10\}\) — B1
    • (b) 9 numbers in the union — M1
    • (b) \(\dfrac{3}{5}\) — A1
  4. 4 Work out [3 marks]

    In a survey of 60 people, 35 own a cat and 28 own a dog. 10 people own neither. Work out the number of people who own both a cat and a dog. [3 marks]

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    Model answer

    \(60 - 10 = 50\) own at least one pet. Then \(35 + 28 - 50 = 13\) own both.

    Mark scheme

    • \(60 - 10 = 50\) — M1
    • \(35 + 28 - 50\) — M1
    • 13 — A1
  5. 5 Work out [4 marks]

    The Venn diagram for 49 people has \(3x\) in set \(A\) only, \(x\) in both sets, \(2x + 1\) in set \(B\) only, and 6 in neither set. (a) Work out the value of \(x\). [2 marks] (b) Work out the probability that a person chosen at random is in both sets. [2 marks]

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    Model answer

    (a) \(3x + x + 2x + 1 + 6 = 49\), so \(6x + 7 = 49\) and \(x = 7\). (b) \(\dfrac{7}{49} = \dfrac{1}{7}\).

    Mark scheme

    • (a) \(6x + 7 = 49\) — M1
    • (a) \(x = 7\) — A1
    • (b) \(\dfrac{7}{49}\) — M1
    • (b) \(\dfrac{1}{7}\) — A1
  6. 6 Work out [3 marks]

    \(P(A) = 0.6\), \(P(B) = 0.5\) and \(P(A \cap B) = 0.3\). (a) Work out \(P(A \cup B)\). [2 marks] (b) Work out the probability that neither \(A\) nor \(B\) happens. [1 mark]

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    Model answer

    (a) \(0.6 + 0.5 - 0.3 = 0.8\). (b) \(1 - 0.8 = 0.2\).

    Mark scheme

    • (a) \(0.6 + 0.5 - 0.3\) — M1
    • (a) \(0.8\) — A1
    • (b) \(0.2\) — B1

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: \(22\)

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