Cambridge O Level Statistics: Probability basics and mutually exclusive events (4040) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover topic 11, Elementary ideas of probability, in the Cambridge O Level Statistics 4040 syllabus for examination in 2025, 2026 and 2027, with the focus on basic probability and mutually exclusive events. Methods are in the study guide and revision notes. The course pages are the O Level Statistics hub and the printable checklist. Show your working in full.

Questions

1. A fair spinner has eight equal sections numbered 1 to 8. Find the probability that it lands on (a) a prime number [1] (b) a multiple of 3 [1] (c) a number that is not a square number [1]

2. A bag contains 7 yellow, 5 green and 3 purple counters. A counter is drawn without looking. Find the probability that it is (a) green [1] (b) yellow or purple [2] (c) not yellow [1]

3. A die is rolled 300 times and shows a six on 66 of the rolls. (a) Estimate the probability of a six with this die. [1] (b) Estimate how many sixes you would expect in 1000 rolls. [2] (c) For a fair die you would expect about 50 sixes in 300 rolls. Does the result prove the die is biased? Give a reason and say what you would do next. [2]

4. A delivery company records the arrival of each parcel as early, on time, late or cancelled. These are mutually exclusive and cover every case. P(early) = 0.15, P(on time) = x, P(late) = 2x and P(cancelled) = 0.1. (a) Find x. [3] (b) Find the probability that a parcel is late or cancelled. [1] (c) Find the probability that a parcel is not early. [1]

5. One card is drawn from each of two piles. Pile A has cards 1, 2, 3. Pile B has cards 1, 2, 3, 4. The score is the sum of the two cards. (a) Draw a table showing all the possible scores. [2] (b) Find P(score is 5). [1] (c) Find P(score is at least 6). [1] (d) Find P(score is even). [2]

6. Of 40 pupils in a club, 22 play chess, 15 take part in drama and 9 do neither. (a) Find the number who do both activities. [2] (b) Find the number who play chess only and the number who do drama only. [1] (c) A pupil is chosen at random. Find the probability that the pupil does exactly one of the activities. [2] (d) Find the probability that the pupil does both. [1]

7. 120 residents are classified by age group and whether they own a car.

	Own a car	No car	Total
Under 30	18	22	40
30 to 59	36	14	50
60 and over	20	10	30
Total	74	46	120

A resident is chosen at random. (a) Find P(60 and over). [1] (b) Find P(owns a car). [1] (c) Find P(under 30 or owns a car). [3] (d) Show that “under 30” and “60 and over” are mutually exclusive, and find P(under 30 or 60 and over). [2]

8. A basketball player scores each free throw with probability 0.7, independently of other throws. She takes two throws. (a) Draw a tree diagram with the probability on each branch. [2] (b) Find the probability that she scores exactly one throw. [2] (c) Find the probability that she scores at least one throw, using the complement. [2]

9. Events A and B have P(A) = 0.52, P(B) = 0.28 and P(A ∪ B) = 0.68. (a) Are A and B mutually exclusive? Explain. [2] (b) Find P(A ∩ B). [2] (c) Find P(neither A nor B). [1] (d) Find P(A but not B). [1]

10. A blood service records the group of 400 donors.

Group	O	A	B	AB
Donors	168	140	68	24

(a) Use the table to estimate the probability that the next donor has group O. [1] (b) Estimate the probability that the next donor has group A or group B. [2] (c) Estimate the probability that the next donor does not have group O. [1] (d) In a drive with 1500 donors, how many would you expect to have group AB? [2]

11. A card is chosen at random from cards numbered 1 to 20. Event A is “a multiple of 4”, event B is “a prime number” and event C is “a square number”. (a) For each pair of events A and B, B and C, A and C, state whether they are mutually exclusive, giving a reason. [3] (b) Find P(A or B). [2] (c) Find P(A or C). [2] (d) Find the probability that the card is none of A, B or C. [2]

12. A spinner lands on red with probability 0.4, blue with probability 0.35 and green with probability 0.25. It is spun twice, and the spins are independent. (a) Find the probability that both spins give the same colour. [3] (b) Find the probability that the colours are different. [1] (c) Find the probability that at least one spin is red. [2] (d) Find the probability that the spins are one blue and one red, in either order. [2]

Answers

1. (a) Primes are 2, 3, 5, 7, so P = 4/8 = 1/2 [1] (b) Multiples of 3 are 3 and 6, so P = 2/8 = 1/4 [1] (c) Squares are 1 and 4, so P(not square) = 1 − 2/8 = 3/4 [1] [3] Examiner insight: Listing the favourable numbers shows the sample space you used; cancel fractions to lowest terms.

2. (a) n(S) = 7 + 5 + 3 = 15, so P(green) = 1/3 [1] (b) Yellow and purple cannot both happen: 7/15 + 3/15 [1] = 10/15 = 2/3 [1] (c) P(not yellow) = 1 − 7/15 = 8/15 [1] [4] Examiner insight: In (b) say you are adding because the colours are mutually exclusive.

3. (a) 66/300 = 0.22 [1] (b) Expected = 0.22 × 1000 [1] = 220 [1] (c) 66 is more than 50, but a die can vary from 50 by chance in 300 rolls [1]. It does not prove bias; roll many more times and see whether the relative frequency stays near 0.22 rather than 1/6 [1]. [5] Examiner insight: Keep the conclusion within the evidence (“suggests”, “not proved”) and add a follow-up action.

4. (a) The probabilities add to 1: 0.15 + x + 2x + 0.1 = 1 [1]; 3x = 0.75 [1]; x = 0.25 [1] (b) P(late or cancelled) = 0.5 + 0.1 = 0.6 [1] (c) P(not early) = 1 − 0.15 = 0.85 [1] [5] Examiner insight: Write the equation “sum of probabilities = 1” before solving for x.

5. (a) Sums (rows: pile A, columns: pile B): [2]

     1  2  3  4
  1  2  3  4  5
  2  3  4  5  6
  3  4  5  6  7

(b) Score 5 appears 3 times out of 12: P = 1/4 [1] (c) Scores of 6 or more: 6, 6, 7 gives 3 outcomes, so P = 3/12 = 1/4 [1] (d) Even scores: 2 (once), 4 (three times), 6 (twice) gives 6 outcomes [1], so P = 6/12 = 1/2 [1] [6] Examiner insight: A complete 12-cell table makes parts (b) to (d) easy to check.

6. (a) At least one activity: 40 − 9 = 31 [1]. Both = 22 + 15 − 31 = 6 [1] (b) Chess only = 22 − 6 = 16; drama only = 15 − 6 = 9 [1] (c) Exactly one = 16 + 9 = 25, so P = 25/40 = 5/8 [1] [1] (d) P(both) = 6/40 = 3/20 [1] [6] Examiner insight: Draw the Venn diagram: the regions 16, 6, 9 and 9 outside add to 40, a quick self-check.

7. (a) P = 30/120 = 1/4 [1] (b) P = 74/120 = 37/60 [1] (c) Under 30: 40/120; owns car: 74/120; both: 18/120 [1]. P = 40/120 + 74/120 − 18/120 [1] = 96/120 = 4/5 [1] (d) No resident is in both age groups, so the events cannot happen together [1]. P = 40/120 + 30/120 = 70/120 = 7/12 [1] [7] Examiner insight: In (c) write the overlap cell (18) down so you remember to subtract it.

8. (a) Two stages, each with branches “score” (0.7) and “miss” (0.3) [1], giving four paths: SS, SM, MS, MM with probabilities 0.49, 0.21, 0.21, 0.09 [1] (b) Exactly one: SM or MS are mutually exclusive paths, 0.21 + 0.21 [1] = 0.42 [1] (c) P(none) = 0.3 × 0.3 = 0.09 [1]; P(at least one) = 1 − 0.09 = 0.91 [1] [6] Examiner insight: Label each branch and check the four path probabilities add to 1.

9. (a) If exclusive, P(A ∪ B) would be 0.52 + 0.28 = 0.80 [1]. This is not 0.68, so they are not mutually exclusive [1] (b) 0.68 = 0.52 + 0.28 − P(A ∩ B) [1], so P(A ∩ B) = 0.12 [1] (c) P(neither) = 1 − 0.68 = 0.32 [1] (d) P(A but not B) = 0.52 − 0.12 = 0.40 [1] [6] Examiner insight: Quote the two figures (0.80 and 0.68) rather than only writing “they overlap”.

10. (a) 168/400 = 0.42 [1] (b) Groups A and B are exclusive: (140 + 68)/400 [1] = 208/400 = 0.52 [1] (c) 1 − 0.42 = 0.58 [1] (d) P(AB) = 24/400 = 0.06 [1]; expected = 0.06 × 1500 = 90 [1] [6] Examiner insight: Do not confuse “group A or group B” with the separate group called AB.

11. (a) A = {4, 8, 12, 16, 20}, B = {2, 3, 5, 7, 11, 13, 17, 19}, C = {1, 4, 9, 16}. A and B share no number, so they are exclusive [1]. B and C share none (1 is not prime), so exclusive [1]. A and C share 4 and 16, so not exclusive [1] (b) P(A or B) = 5/20 + 8/20 [1] = 13/20 [1] (c) P(A or C) = 5/20 + 4/20 − 2/20 [1] = 7/20 [1] (d) Numbers in A, B or C: 7 + 8 = 15 [1]. None: 5 cards, so P = 5/20 = 1/4 [1] [9] Examiner insight: For each pair, name an overlapping number or state that none exists.

12. (a) P(same colour) = P(RR) + P(BB) + P(GG) [1] = 0.16 + 0.1225 + 0.0625 [1] = 0.345 [1] (b) 1 − 0.345 = 0.655 [1] (c) P(no red) = 0.6 × 0.6 = 0.36 [1]; P(at least one red) = 1 − 0.36 = 0.64 [1] (d) P(BR) + P(RB) = 0.35 × 0.4 + 0.4 × 0.35 [1] = 0.28 [1] [8] Examiner insight: In (d) write both orders as separate paths.

Where marks are usually lost

• Not listing the sample space fully, so outcomes are missed or counted twice.
• Counting an overlap twice when events can happen together, as in question 11(c).
• Forgetting to simplify, or to check that a probability lies between 0 and 1.
• Treating “A or B” as “A and B”, or reading group AB as “A or B”.
• Not showing the rule or the sum, so a slip cannot be traced.

Next steps

• Recap with the revision notes and the study guide.
• Back to the O Level Statistics hub and the printable checklist.
• Take the free 10-minute diagnostics to find weak spots.
• Book a free trial class for help with anything you got wrong.

Official syllabus

Cambridge O Level Statistics 4040, syllabus for examination in 2025, 2026 and 2027, published by Cambridge University Press & Assessment (Cambridge International Education).