OpenRevise

Exam questions · Maths · Probability

Probability Basics and Relative Frequency

  • 6 exam questions
  • 16 marks
  • 9 quick checks
  1. 1 Find [2 marks]

    The letters of the word MATHEMATICS are written on 11 cards, one letter on each card. One card is chosen at random. Find the probability that it is (a) the letter M, (b) a vowel. [2 marks]

    Show answerHide answer

    Model answer

    (a) There are two Ms, so \(\dfrac{2}{11}\). (b) The vowels are A, E, A, I, so \(\dfrac{4}{11}\).

    Mark scheme

    • (a) \(\dfrac{2}{11}\) — B1
    • (b) \(\dfrac{4}{11}\) — B1
  2. 2 Calculate [3 marks]

    Hana spins a spinner 100 times. The results are red 38, blue 27 and green 35. (a) Write down the relative frequency of blue as a decimal. [1 mark] (b) Hana spins the spinner 500 times. Estimate the number of times it lands on blue. [2 marks]

    Show answerHide answer

    Model answer

    (a) \(\dfrac{27}{100} = 0.27\). (b) \(0.27 \times 500 = 135\).

    Mark scheme

    • (a) \(0.27\) — B1
    • (b) \(0.27 \times 500\) — M1
    • (b) 135 — A1
  3. 3 Calculate [3 marks]

    The probability that a bus is on time is 0.72. (a) Calculate the probability that the bus is not on time. [1 mark] (b) Out of 50 days, estimate the number of days that the bus is on time. [2 marks]

    Show answerHide answer

    Model answer

    (a) \(1 - 0.72 = 0.28\). (b) \(0.72 \times 50 = 36\).

    Mark scheme

    • (a) \(0.28\) — B1
    • (b) \(0.72 \times 50\) — M1
    • (b) 36 — A1
  4. 4 Calculate [3 marks]

    A bag contains only red, white and green counters. The probability of picking a red counter is \(x\). The probability of picking a white counter is \(2x\). The probability of picking a green counter is 0.4. (a) Calculate the value of \(x\). [2 marks] (b) Write down the probability of picking a white counter. [1 mark]

    Show answerHide answer

    Model answer

    (a) \(x + 2x + 0.4 = 1\), so \(3x = 0.6\) and \(x = 0.2\). (b) \(2x = 0.4\).

    Mark scheme

    • (a) \(x + 2x + 0.4 = 1\) — M1
    • (a) \(0.2\) — A1
    • (b) \(0.4\) — B1
  5. 5 Explain [2 marks]

    Ben throws a normal dice. He says, “Either I get a six or I do not, so the probability of a six is \(\dfrac{1}{2}\).” Explain why Ben is wrong. [2 marks]

    Show answerHide answer

    Model answer

    The two outcomes are not equally likely. There is only one way to get a six out of six equally likely outcomes, so the probability is \(\dfrac{1}{6}\).

    Mark scheme

    • States that the outcomes are not equally likely — B1
    • States that the probability of a six is \(\dfrac{1}{6}\) — B1
  6. 6 Calculate [3 marks]

    \(P(F) = 0.55\), \(P(T) = 0.35\) and the probability that neither \(F\) nor \(T\) happens is 0.2. Calculate \(P(F \text{ and } T)\). [3 marks]

    Show answerHide answer

    Model answer

    \(P(F \text{ or } T) = 1 - 0.2 = 0.8\). Using \(P(F \text{ or } T) = P(F) + P(T) - P(F \text{ and } T)\) gives \(0.8 = 0.9 - P(F \text{ and } T)\), so \(P(F \text{ and } T) = 0.1\).

    Mark scheme

    • \(1 - 0.2 = 0.8\) — M1
    • \(0.55 + 0.35 - 0.8\) — M1
    • \(0.1\) — 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\(-1\)
    Show answerHide answer

    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\(\dfrac{3}{5}\)
    Show answerHide answer

    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\(\dfrac{4}{5}\)
    Show answerHide answer

    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\(4\)
    Show answerHide answer

    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\(600\)
    Show answerHide answer

    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\(\dfrac{1}{36}\)
    Show answerHide answer

    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\(\dfrac{1}{2}\)
    Show answerHide answer

    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\(200\)
    Show answerHide answer

    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\(\dfrac{5}{26}\)
    Show answerHide answer

    B: \(\dfrac{4}{13}\)

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