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Exam questions · Maths · Probability

Probability Basics and Relative Frequency

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

    Ten cards are numbered 1 to 10. One card is picked at random. Work out the probability that the card shows (a) a prime number, (b) a multiple of 3. [2 marks]

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    Model answer

    (a) The primes are 2, 3, 5 and 7, so \(\dfrac{4}{10} = \dfrac{2}{5}\). (b) The multiples of 3 are 3, 6 and 9, so \(\dfrac{3}{10}\).

    Mark scheme

    • (a) \(\dfrac{4}{10}\) or \(\dfrac{2}{5}\) or 0.4 — B1
    • (b) \(\dfrac{3}{10}\) or 0.3 — B1
  2. 2 Choose [3 marks]

    Here are five words: impossible, unlikely, even chance, likely, certain. Choose the best word to describe each event. (a) A fair coin lands on heads. [1 mark] (b) A normal dice lands on 7. [1 mark] (c) A fair dice lands on a number less than 6. [1 mark]

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    Model answer

    (a) Even chance. (b) Impossible. (c) Likely, because the probability is \(\dfrac{5}{6}\).

    Mark scheme

    • (a) Even chance — B1
    • (b) Impossible — B1
    • (c) Likely — B1
  3. 3 Work out [4 marks]

    Jack spins a spinner 80 times. It lands on green 28 times. (a) Work out the relative frequency of green. Give your answer as a fraction in its simplest form. [2 marks] (b) Jack spins the spinner 200 more times. Estimate the number of times it lands on green. [2 marks]

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    Model answer

    (a) \(\dfrac{28}{80} = \dfrac{7}{20}\). (b) \(\dfrac{7}{20} \times 200 = 70\).

    Mark scheme

    • (a) \(\dfrac{28}{80}\) — M1
    • (a) \(\dfrac{7}{20}\) — A1
    • (b) \(\dfrac{7}{20} \times 200\) — M1
    • (b) 70 — A1
  4. 4 Work out [3 marks]

    A bag contains only red, blue and yellow counters. The probability of picking a red counter is 0.3. The probability of picking a blue counter is 0.45. There are 40 counters in the bag. Work out the number of yellow counters. [3 marks]

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    Model answer

    \(P(\text{yellow}) = 1 - 0.3 - 0.45 = 0.25\). Then \(0.25 \times 40 = 10\) yellow counters.

    Mark scheme

    • \(1 - 0.3 - 0.45\) — M1
    • \(0.25\) — A1
    • 10 — A1
  5. 5 Work out [2 marks]

    The probability that a seed grows is 0.9. Ella plants 300 seeds. Work out an estimate for the number of seeds that grow. [2 marks]

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    Model answer

    \(0.9 \times 300 = 270\).

    Mark scheme

    • \(0.9 \times 300\) — M1
    • 270 — A1
  6. 6 Work out [3 marks]

    \(A\) and \(B\) are events. \(P(A) = 0.5\), \(P(B) = 0.4\) and \(P(A \text{ or } B) = 0.7\). Work out \(P(A \text{ and } B)\). [3 marks]

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    Model answer

    \(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\), so \(0.7 = 0.9 - P(A \text{ and } B)\). Then \(P(A \text{ and } B) = 0.9 - 0.7 = 0.2\).

    Mark scheme

    • \(0.5 + 0.4 = 0.9\) — M1
    • \(0.9 - 0.7\) — M1
    • \(0.2\) — A1

Quick check

  1. 1

    What is the probability of an impossible event?

    1. A\(1\)
    2. B\(0\)
    3. C\(\dfrac{1}{2}\)
    4. D\(-1\)
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    B: \(0\)

    An impossible event has probability 0.

  2. 2

    A bag has 3 red, 5 blue and 2 green counters. One is taken at random. What is the probability it is blue?

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

    There are 10 counters and 5 are blue, so \(\dfrac{5}{10} = \dfrac{1}{2}\).

  3. 3

    A bag has 3 red, 5 blue and 2 green counters. What is the probability that a counter taken at random is not green?

    1. A\(\dfrac{1}{5}\)
    2. B\(\dfrac{2}{5}\)
    3. C\(\dfrac{3}{10}\)
    4. D\(\dfrac{4}{5}\)
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    D: \(\dfrac{4}{5}\)

    \(1 - \dfrac{2}{10} = \dfrac{8}{10} = \dfrac{4}{5}\).

  4. 4

    A dice is thrown 120 times and a 6 comes up 30 times. What is the relative frequency of a 6?

    1. A\(\dfrac{1}{6}\)
    2. B\(\dfrac{1}{30}\)
    3. C\(\dfrac{1}{4}\)
    4. D\(4\)
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    C: \(\dfrac{1}{4}\)

    \(\dfrac{30}{120} = \dfrac{1}{4}\).

  5. 5

    A fair dice is thrown 600 times. How many sixes are expected?

    1. A\(60\)
    2. B\(100\)
    3. C\(120\)
    4. D\(600\)
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    B: \(100\)

    \(\dfrac{1}{6} \times 600 = 100\).

  6. 6

    Two fair dice are thrown. What is the probability that the total score is 7?

    1. A\(\dfrac{1}{6}\)
    2. B\(\dfrac{1}{12}\)
    3. C\(\dfrac{7}{36}\)
    4. D\(\dfrac{1}{36}\)
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    A: \(\dfrac{1}{6}\)

    There are 6 ways to make 7 out of 36, so \(\dfrac{6}{36} = \dfrac{1}{6}\).

  7. 7

    A bag has 3 red, 5 blue and 2 green counters. What is the probability of taking a red or a green counter?

    1. A\(\dfrac{3}{5}\)
    2. B\(\dfrac{1}{5}\)
    3. C\(\dfrac{3}{10}\)
    4. D\(\dfrac{1}{2}\)
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    D: \(\dfrac{1}{2}\)

    The events are mutually exclusive, so \(\dfrac{3}{10} + \dfrac{2}{10} = \dfrac{1}{2}\).

  8. 8

    A spinner is spun 200 times and lands on red 74 times. How many reds would be expected if the probability of red were \(\dfrac{1}{4}\)?

    1. A\(74\)
    2. B\(25\)
    3. C\(50\)
    4. D\(200\)
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    C: \(50\)

    \(\dfrac{1}{4} \times 200 = 50\). The result of 74 suggests the spinner may be biased.

  9. 9

    A card is taken from a pack of 52. What is the probability that it is a heart or a king?

    1. A\(\dfrac{17}{52}\)
    2. B\(\dfrac{4}{13}\)
    3. C\(\dfrac{1}{13}\)
    4. D\(\dfrac{5}{26}\)
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    B: \(\dfrac{4}{13}\)

    \(\dfrac{13}{52} + \dfrac{4}{52} - \dfrac{1}{52} = \dfrac{16}{52} = \dfrac{4}{13}\).