Probability

50 maths questions with worked answers with answers for P1–P9 Probability in Edexcel GCSE Maths (spec 1MA1). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official Pearson material.

1. Question 1 · Relative frequency · P1

   A drawing pin lands point up 36 times in 120 throws. Estimate the probability it lands point up.

   Answer:

   Relative frequency = 36120.
   0.3.

2. Question 2 · Relative frequency · P1

   A biased dice shows a six 45 times in 300 rolls. Estimate how many sixes in 1000 rolls.

   Answer:

   Relative frequency = 45300 = 0.15.
   0.15 × 1000 = 150.

3. Question 3 · Fairness · P2

   A coin lands heads 7 times out of 10. Is it biased? Explain.

   Answer:

   10 trials is too few to decide.
   More trials are needed; relative frequency gets more reliable as trials increase.

4. Question 4 · Probability scale · P3

   An event has probability 0. What does this mean?

   Answer:

   It is impossible.
   An event with probability 1 is certain.

5. Question 5 · Basic probability · P3

   A bag has 4 red, 5 blue and 3 green counters. Find P(blue).

   Answer:

   12 counters in total.
   512.

6. Question 6 · Basic probability · P3

   A fair dice is rolled. Find P(a number greater than 4).

   Answer:

   5 or 6: 2 outcomes out of 6.
   26 = 13.

7. Question 7 · Basic probability · P3

   A card is picked from 1–20. Find P(a multiple of 3).

   Answer:

   3, 6, 9, 12, 15, 18: 6 outcomes.
   620 = 310.

8. Question 8 · Sum to 1 · P4

   P(rain) = 0.35. Find P(not rain).

   Answer:

   P(not A) = 1 − P(A).
   1 − 0.35 = 0.65.

9. Question 9 · Sum to 1 · P4

   A spinner lands on red, blue or green. P(red) = 0.2 and P(blue) = 0.45. Find P(green).

   Answer:

   Probabilities sum to 1.
   1 − 0.65 = 0.35.

10. Question 10 · Sum to 1 · P4

    A spinner has P(A) = x, P(B) = 2x and P(C) = 0.4. Find x.

    Answer:

    x + 2x + 0.4 = 1, so 3x = 0.6.
    x = 0.2.

11. Question 11 · Expected outcomes · P3

    P(a train is late) = 0.08. How many of 250 trains are expected to be late?

    Answer:

    Expected = probability × number of trials.
    0.08 × 250 = 20.

12. Question 12 · Expected outcomes · P3

    A fair dice is rolled 90 times. How many fours are expected?

    Answer:

    P(4) = 16.
    90 ÷ 6 = 15.

13. Question 13 · Sample spaces · P7

    Two coins are flipped. List the sample space.

    Answer:

    Each coin is H or T.
    HH, HT, TH, TT.

14. Question 14 · Sample spaces · P7

    Two fair dice are rolled and the scores added. Find P(total of 7).

    Answer:

    There are 36 outcomes; 6 give 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
    636 = 16.

15. Question 15 · Sample spaces · P7

    Two fair dice are rolled. Find P(double).

    Answer:

    6 doubles out of 36 outcomes.
    16.

16. Question 16 · Venn diagrams · P6

    What does A ∩ B mean?

    Answer:

    The intersection.
    Outcomes in both A and B.

17. Question 17 · Venn diagrams · P6

    What does A ∪ B mean?

    Answer:

    The union.
    Outcomes in A or B or both.

18. Question 18 · Venn diagrams · P6

    What does A′ mean?

    Answer:

    The complement of A.
    Outcomes not in A.

19. Question 19 · Venn diagrams · P6

    ξ = {1, 2, …, 10}, A = even numbers, B = multiples of 3. List A ∩ B.

    Answer:

    Even multiples of 3 up to 10.
    {6}.

20. Question 20 · Venn diagrams · P6

    In a class of 30, 18 play football, 12 play tennis and 5 play both. How many play neither?

    Answer:

    Football or tennis = 18 + 12 − 5 = 25.
    30 − 25 = 5.

21. Question 21 · Venn diagrams · P6

    Using the class above (30 pupils, 18 football, 12 tennis, 5 both), find P(tennis only).

    Answer:

    Tennis only = 12 − 5 = 7.
    730.

22. Question 22 · Mutually exclusive · P4

    What are mutually exclusive events?

    Answer:

    Events that cannot happen at the same time.
    P(A or B) = P(A) + P(B).

23. Question 23 · Mutually exclusive · P4

    P(red) = 0.3 and P(blue) = 0.25 for one spin. Find P(red or blue).

    Answer:

    They are mutually exclusive.
    0.3 + 0.25 = 0.55.

24. Question 24 · Independent events · P8

    What are independent events?

    Answer:

    One event does not affect the probability of the other.
    P(A and B) = P(A) × P(B).

25. Question 25 · Independent events · P8

    A coin is flipped and a dice is rolled. Find P(head and a 6).

    Answer:

    12 × 16.
    112.

26. Question 26 · Tree diagrams · P8

    P(pass driving test) = 0.7. Two friends take it independently. Find P(both pass).

    Answer:

    0.7 × 0.7.
    0.49.

27. Question 27 · Tree diagrams · P8

    P(pass) = 0.7 for each of two independent friends. Find P(exactly one passes).

    Answer:

    P(pass, fail) + P(fail, pass) = 0.7 × 0.3 + 0.3 × 0.7.
    0.21 + 0.21 = 0.42.

28. Question 28 · Tree diagrams · P8

    P(pass) = 0.7 for each of two friends. Find P(at least one passes).

    Answer:

    P(at least one) = 1 − P(neither).
    1 − 0.3 × 0.3 = 1 − 0.09 = 0.91.

29. Question 29 · Tree diagrams · P8

    Why do branches from one point on a tree diagram add up to 1?

    Answer:

    They show every possible outcome at that stage.
    The outcomes are mutually exclusive and exhaustive.

30. Question 30 · Without replacement · P8

    A bag has 5 red and 3 blue balls. Two are taken without replacement. Find P(both red).

    Answer:

    58 × 47.
    2056 = 514.

31. Question 31 · Without replacement · P8

    A bag has 5 red and 3 blue balls. Two are taken without replacement. Find P(one of each colour).

    Answer:

    P(R, B) + P(B, R) = 58 × 37 + 38 × 57.
    3056 = 1528.

32. Question 32 · Without replacement · P8

    A box has 4 milk and 6 dark chocolates. Two are eaten. Find P(both dark).

    Answer:

    610 × 59.
    3090 = 13.

33. Question 33 · Conditional probability · P9 · Higher only

    What is conditional probability?

    Answer:

    The probability of an event given that another has happened.
    Written P(A | B).

34. Question 34 · Conditional probability · P9 · Higher only

    Of 40 pupils, 25 have a dog. 10 of the dog owners also have a cat. A dog owner is chosen. Find P(has a cat).

    Answer:

    Restrict to the 25 dog owners.
    1025 = 25.

35. Question 35 · Conditional probability · P9 · Higher only

    P(A) = 0.6, P(B) = 0.5 and P(A ∩ B) = 0.3. Find P(A | B).

    Answer:

    P(A | B) = P(A ∩ B)P(B).
    0.30.5 = 0.6.

36. Question 36 · Independence test · P9 · Higher only

    P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2. Are A and B independent?

    Answer:

    P(A) × P(B) = 0.4 × 0.5 = 0.2.
    This equals P(A ∩ B), so yes.

37. Question 37 · Without replacement · P8

    There are 3 red and 2 green sweets. Sam takes two. Show that P(same colour) = 25.

    Answer:

    P(RR) = 35 × 24 = 620. P(GG) = 25 × 14 = 220.
    820 = 25.

38. Question 38 · Two-way tables · P9 · Higher only

    In a two-way table, 12 of the 30 Year 10s and 18 of the 50 Year 11s chose art. A pupil who chose art is picked. Find P(Year 10).

    Answer:

    30 chose art in total.
    1230 = 25.

39. Question 39 · Relative frequency · P5

    Why does more trials give a better estimate of probability?

    Answer:

    Random variation evens out over many trials.
    Relative frequency gets closer to the true probability.

40. Question 40 · Basic probability · P3

    A letter is chosen at random from the word PROBABILITY. Find P(B).

    Answer:

    There are 11 letters and 2 Bs.
    211.

41. Question 41 · Expected outcomes · P3

    A spinner has P(red) = 0.25. It is spun 60 times. How many reds are expected?

    Answer:

    0.25 × 60.
    15.

42. Question 42 · Listing · P6

    Ann, Ben and Cal line up in a queue. List all the orders.

    Answer:

    3 × 2 × 1 = 6 orders.
    ABC, ACB, BAC, BCA, CAB, CBA.

43. Question 43 · Independent events · P8

    A fair dice is rolled twice. Find P(two sixes).

    Answer:

    16 × 16.
    136.

44. Question 44 · Repeated events · P8 · Higher only

    A fair coin is flipped 3 times. Find P(at least one head).

    Answer:

    P(no heads) = (12)³ = 18.
    1 − 18 = 78.

45. Question 45 · Sum to 1 · P4

    P(win) = 38 and draws are impossible. Find P(lose).

    Answer:

    1 − 38.
    58.

46. Question 46 · Frequency trees · P6

    What is a frequency tree used for?

    Answer:

    To sort a number of people or items into groups at each stage.
    It shows frequencies, not probabilities.

47. Question 47 · Basic probability · P3

    A bag has only red and blue counters. P(red) = 27. There are 10 blue counters. How many red?

    Answer:

    P(blue) = 57, so 5 parts = 10 and 1 part = 2.
    4 red counters.

48. Question 48 · Without replacement · P8 · Higher only

    A bag has n balls, 3 of which are red. Two are taken without replacement. Write P(both red) in terms of n.

    Answer:

    3n × 2n − 1.
    6n(n − 1).

49. Question 49 · Equally likely · P2

    Why is P(a 3) = 16 on a fair dice?

    Answer:

    The six outcomes are equally likely.
    One of them is a 3.

50. Question 50 · Tree diagrams · P8

    P(late on Monday) = 0.1 and P(late on Tuesday) = 0.1, independently. Find P(late on neither day).

    Answer:

    P(not late) = 0.9 each day.
    0.9 × 0.9 = 0.81.

More Edexcel GCSE Maths topics

• N1–N9 Structure and calculation
• N10–N12 Fractions, decimals and percentages
• N13–N16 Measures and accuracy
• A1–A7 Notation, vocabulary and manipulation
• A8–A16 Graphs
• A17–A22 Solving equations and inequalities
• A23–A25 Sequences
• R1–R16 Ratio, proportion and rates of change
• G1–G15 Properties and constructions
• G16–G23 Mensuration and calculation
• G24–G25 Vectors
• P1–P9 Probability
• S1–S6 Statistics

Other Maths courses: AQA GCSE Maths, OCR GCSE Maths.