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Exam questions · Maths · Probability

Frequency Trees and Two-Way Tables

  • 6 exam questions
  • 21 marks
  • 9 quick checks
  1. 1 Work out [3 marks]

    There are 60 people at a gym. 36 of them are women. 9 of the women and 6 of the men use the pool. (a) Work out the number of men. (1 mark) (b) Work out the number of people who do not use the pool. (2 marks)

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    Model answer

    (a) \(60 - 36 = 24\). (b) \(9 + 6 = 15\) people use the pool, so \(60 - 15 = 45\) do not.

    Mark scheme

    • (a) 24 — B1
    • (b) \(9 + 6 = 15\) — M1
    • (b) 45 — A1
  2. 2 Work out [4 marks]

    The two-way table shows information about how 100 students travel to school. Some of the numbers are missing. (a) Complete the two-way table. (3 marks) (b) A student is chosen at random. Work out the probability that the student is in Year 11 and walks. (1 mark)

    A two-way table of how Year 10 and Year 11 students travel to school, with four missing values.
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    Model answer

    (a) Year 10 cycle: \(50 - 24 - 18 = 8\). Year 11 bus: \(44 - 24 = 20\). Year 11 total: 50. Total who walk: \(18 + 25 = 43\). (b) \(\dfrac{25}{100} = \dfrac{1}{4}\).

    Mark scheme

    • (a) At least two correct values — B1
    • (a) At least three correct values — B1
    • (a) All four correct: 8, 20, 50 and 43 — B1
    • (b) \(\dfrac{25}{100}\) or \(\dfrac{1}{4}\) — B1
  3. 3 Work out [3 marks]

    A survey asks 200 people if they own a dog. 120 of the people are female, and the rest are male. 30% of the females and 25% of the males own a dog. Work out the total number of people who own a dog.

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    Model answer

    Females: \(0.3 \times 120 = 36\). There are 80 males, and \(0.25 \times 80 = 20\). The total is \(36 + 20 = 56\).

    Mark scheme

    • \(0.3 \times 120 = 36\) — M1
    • \(0.25 \times 80 = 20\) — M1
    • 56 — A1
  4. 4 Work out [3 marks]

    In a school, 60% of the students are girls. 20% of the girls and 40% of the boys play hockey. A student is chosen at random. Work out the probability that the student plays hockey.

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    Model answer

    \(0.6 \times 0.2 + 0.4 \times 0.4 = 0.12 + 0.16 = 0.28\).

    Mark scheme

    • \(0.6 \times 0.2\) or \(0.4 \times 0.4\) — M1
    • \(0.6 \times 0.2 + 0.4 \times 0.4\) — M1
    • \(0.28\) — A1
  5. 5 Work out [4 marks]

    There are 150 students in Year 11. 90 of them are boys. 35 of the boys and 25 of the girls are left-handed. (a) Work out the number of girls. (1 mark) (b) Work out the number of students who are right-handed. (2 marks) (c) A student is chosen at random. Work out the probability that the student is a left-handed girl. (1 mark)

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    Model answer

    (a) \(150 - 90 = 60\). (b) Right-handed boys \(90 - 35 = 55\), right-handed girls \(60 - 25 = 35\), so \(55 + 35 = 90\). (c) \(\dfrac{25}{150} = \dfrac{1}{6}\).

    Mark scheme

    • (a) 60 — B1
    • (b) \(55\) and \(35\) seen, or \(150 - 35 - 25\) — M1
    • (b) 90 — A1
    • (c) \(\dfrac{1}{6}\) or \(\dfrac{25}{150}\) — B1
  6. 6 Work out [4 marks]

    There are 80 pupils on a trip. The ratio of boys to girls is \(3 : 5\). 40% of the boys and 20% of the girls bring a packed lunch. Work out the fraction of all the pupils who bring a packed lunch. Give your answer in its simplest form.

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    Model answer

    Boys: \(80 \div 8 \times 3 = 30\). Girls: 50. Packed lunches: \(0.4 \times 30 = 12\) and \(0.2 \times 50 = 10\), which is 22. The fraction is \(\dfrac{22}{80} = \dfrac{11}{40}\).

    Mark scheme

    • 30 boys and 50 girls — M1
    • \(0.4 \times 30 = 12\) and \(0.2 \times 50 = 10\) — M1
    • 22 — A1
    • \(\dfrac{11}{40}\) — A1

Quick check

  1. 1

    A frequency tree starts with 100 pupils, and 60 are girls. How many are boys?

    1. A\(40\)
    2. B\(60\)
    3. C\(160\)
    4. D\(100\)
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    A: \(40\)

    \(100 - 60 = 40\).

  2. 2

    40% of 60 girls walk to school. How many girls walk?

    1. A\(36\)
    2. B\(40\)
    3. C\(20\)
    4. D\(24\)
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    D: \(24\)

    \(0.4 \times 60 = 24\).

  3. 3

    In a two-way table the Year 10 row total is 50. 24 take the bus and 18 walk. How many cycle?

    1. A\(42\)
    2. B\(6\)
    3. C\(8\)
    4. D\(18\)
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    C: \(8\)

    \(50 - 24 - 18 = 8\).

  4. 4

    In a survey of 100 students, 24 are in Year 10 and take the bus. What is the probability that a student is in Year 10 and takes the bus?

    1. A\(\dfrac{1}{24}\)
    2. B\(\dfrac{6}{25}\)
    3. C\(\dfrac{24}{50}\)
    4. D\(\dfrac{1}{4}\)
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    B: \(\dfrac{6}{25}\)

    \(\dfrac{24}{100} = \dfrac{6}{25}\).

  5. 5

    13 out of 100 students cycle to school. How many of 500 students would you expect to cycle?

    1. A\(65\)
    2. B\(13\)
    3. C\(130\)
    4. D\(6.5\)
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    A: \(65\)

    \(\dfrac{13}{100} \times 500 = 65\).

  6. 6

    What is 25% of 120?

    1. A\(25\)
    2. B\(48\)
    3. C\(95\)
    4. D\(30\)
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    D: \(30\)

    \(0.25 \times 120 = 30\).

  7. 7

    200 people go to a gym. 60% are women. 25% of the women and 40% of the men go in the morning. How many people go in the morning?

    1. A\(54\)
    2. B\(60\)
    3. C\(62\)
    4. D\(70\)
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    C: \(62\)

    120 women and 80 men. \(30 + 32 = 62\).

  8. 8

    50 people were asked about pets. 28 have a cat, 20 have a dog and 6 have both. How many have neither?

    1. A\(6\)
    2. B\(8\)
    3. C\(14\)
    4. D\(22\)
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    B: \(8\)

    At least one: \(22 + 6 + 14 = 42\). \(50 - 42 = 8\).

  9. 9

    In a frequency tree, the branches after “60 girls” show 24 who walk and some who take the bus. How many take the bus?

    1. A\(36\)
    2. B\(24\)
    3. C\(84\)
    4. D\(60\)
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    A: \(36\)

    The branches add up to the number they came from: \(60 - 24 = 36\).