Cambridge IGCSE Statistics: Exclusive and independent events, conditional probability, tree diagrams (0479) – 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.

The questions below test outcomes 7.5, 7.6 and 7.7 of Topic 7 (Probability) in the 2027 Cambridge IGCSE Statistics (0479) syllabus. Start with the study guide first, or use the revision notes to refresh the rules. Links: course hub, checklist and free diagnostics.

Questions

1. A customer rating is recorded as poor, fair, good or excellent. P(poor) = 0.08, P(fair) = 0.22 and P(good) = 0.45.

(a) Find P(excellent). [1] (b) Find the probability that a rating is fair or good, and explain why you may add. [2]

2. A and B are independent events with P(A) = 0.35 and P(B) = 0.6.

(a) Find P(A ∩ B). [1] (b) Find P(A ∪ B). [2] (c) Find the probability that A and B both fail to occur. [1]

3. In a sample of 160 adults, 96 use a fitness app (A), 40 are aged over 50 (B) and 24 are both.

(a) Show that A and B are independent. [3] (b) In a second sample of 160 adults, the counts for A and B are the same, but 30 are both. Explain whether A and B are independent in this sample. [2]

4. Events C and D are mutually exclusive with P(C) = 0.5 and P(D) = 0.2.

(a) Find P(C ∪ D). [1] (b) Explain why C and D cannot be independent. [2]

5. A gym has 90 members. The table shows their usual session time and type of membership.

	Adult	Student
Morning	24	11
Evening	31	24

A member is chosen at random.

(a) Find the probability that the member is a student. [1] (b) Given that the member attends in the evening, find the probability they are a student. [2] (c) Given that the member is a student, find the probability they attend in the evening. [2]

6. A box holds 15 chocolates: 6 dark, 5 milk and 4 white. Chocolates are taken at random and not replaced. The first chocolate taken is dark.

(a) Find the probability that the second chocolate is also dark. [2] (b) Find the probability that the second chocolate is not milk. [2]

7. To gain a licence, a candidate takes a theory test and a practical test. The probability of passing the theory test is 0.75 and of passing the practical test is 0.6. The results are independent.

(a) Draw a tree diagram showing all four outcomes and their probabilities. [2] (b) Find the probability of passing both tests. [1] (c) Find the probability of passing exactly one test. [2] (d) Find the probability of passing at least one test. [1]

8. A bag contains 8 tiles: 3 show a vowel and 5 show a consonant. Two tiles are taken at random without replacement.

(a) Draw a tree diagram with the probability on every branch. [2] (b) Find the probability that both tiles show consonants. [1] (c) Find the probability that one shows a vowel and one a consonant. [2] (d) Find the probability that at least one shows a vowel. [1]

9. Machine A makes 60% of the phones in a factory and machine B makes the rest. 3% of phones from A are defective and 8% of phones from B are defective. One phone is chosen at random.

(a) Draw a tree diagram for this information. [2] (b) Find the probability that the phone is defective. [2] (c) The phone is found to be defective. Find the probability it came from machine B. [3]

10. In each of three rounds of a quiz game, a player answers correctly with probability 0.6, independently of the other rounds.

(a) Find the probability that she answers all three rounds correctly. [1] (b) Find the probability that she answers exactly two rounds correctly. [2] (c) Find the probability that she answers at least one round correctly. [2] (d) To qualify she needs at least two correct answers. Given that she qualifies, find the probability that she answered all three correctly. [3]

11. Bag P holds 4 red and 6 white balls. Bag Q holds 5 red and 3 white balls. A fair coin is tossed. If it lands heads, a ball is taken at random from bag P. If it lands tails, a ball is taken at random from bag Q.

(a) Draw a tree diagram for this experiment. [2] (b) Find the probability that the ball is red. [2] (c) The ball is red. Find the probability that it came from bag P. [3] (d) By comparing P(heads and red) with P(heads) × P(red), state whether “the coin lands heads” and “the ball is red” are independent. [2]

Answers

1. (a) 1 - 0.08 - 0.22 - 0.45 = 0.25 [1] (b) The ratings fair and good cannot both be recorded for one customer, so they are mutually exclusive and the probabilities add. [1] 0.22 + 0.45 = 0.67 [1] Examiner insight: Naming the reason for adding (a customer cannot give two ratings) is advice worth following whenever a question says “explain why”, because the answer then shows the idea and not just the sum.

2. (a) P(A ∩ B) = 0.35 × 0.6 = 0.21 [1] (b) P(A ∪ B) = 0.35 + 0.6 - 0.21 [1] = 0.74 [1] (c) 1 - 0.74 = 0.26 [1] Examiner insight: Writing the subtraction of the overlap as a separate line makes the method clear.

3. (a) P(A) = 96/160 = 0.6 and P(B) = 40/160 = 0.25 [1] P(A ∩ B) = 24/160 = 0.15 [1] P(A) × P(B) = 0.6 × 0.25 = 0.15, which equals P(A ∩ B), so A and B are independent [1] (b) P(A ∩ B) = 30/160 = 0.1875 [1] This differs from 0.6 × 0.25 = 0.15, so A and B are not independent in this sample [1] Examiner insight: For a “show that” question, state all three probabilities, the product, and the comparison. Stopping at the numbers without a conclusion leaves the argument unfinished.

4. (a) 0.5 + 0.2 = 0.7 [1] (b) Mutually exclusive events cannot both happen, so P(C ∩ D) = 0 [1] For independence it would need to be 0.5 × 0.2 = 0.1, which is not 0, so C and D are not independent [1] Examiner insight: A convincing answer gives both numbers, 0 and 0.1, and compares them.

5. (a) 35/90 = 7/18 [1] (b) The evening group has 31 + 24 = 55 members [1] P(student | evening) = 24/55 [1] So 24/55 (c) The student group has 11 + 24 = 35 members [1] P(evening | student) = 24/35 [1] So 24/35 Examiner insight: Underline the word after “given” and use that row or column total as the denominator. Using 90 for every part is the usual loss in these questions.

6. (a) After one dark chocolate is removed, 14 remain, 5 of them dark [1] P(dark) = 5/14 [1] (b) 5 milk of 14 remain, so not milk is 14 - 5 = 9 out of 14 [1] 9/14 [1] Examiner insight: Stating the new total and the new count (14 and 5) shows the idea of conditional probability and is the working worth setting out clearly.

7. (a) Theory branches 0.75 pass and 0.25 fail [1] From each, practical branches 0.6 pass and 0.4 fail, with the four products 0.45, 0.30, 0.15 and 0.10 [1]

Theory         Practical       Outcome        Probability
               0.6 -- Pass     Pass, Pass     0.45
   0.75 -- Pass
               0.4 -- Fail     Pass, Fail     0.30
Start
               0.6 -- Pass     Fail, Pass     0.15
   0.25 -- Fail
               0.4 -- Fail     Fail, Fail     0.10

(b) 0.75 × 0.6 = 0.45 [1] (c) 0.75 × 0.4 + 0.25 × 0.6 [1] = 0.30 + 0.15 = 0.45 [1] (d) 1 - 0.25 × 0.4 = 1 - 0.1 = 0.9 [1] Examiner insight: Label every branch with its probability and check that each pair adds to 1. A clearly labelled tree also makes your later working easier to follow.

8. (a) First stage: vowel 3/8, consonant 5/8 [1] Second stage: after a vowel, vowel 2/7 and consonant 5/7; after a consonant, vowel 3/7 and consonant 4/7 [1] (b) 5/8 × 4/7 = 20/56 = 5/14 [1] (c) 3/8 × 5/7 + 5/8 × 3/7 [1] = 15/56 + 15/56 = 30/56 = 15/28 [1] (d) 1 - 5/14 = 9/14 [1] Examiner insight: The denominators at the second stage must be 7. Check that the four outcomes sum to 1: 6/56 + 15/56 + 15/56 + 20/56 = 56/56.

9. (a) First stage: A with 0.6 and B with 0.4 [1] Second stage: A then defective 0.03 and not defective 0.97; B then defective 0.08 and not defective 0.92 [1] (b) 0.6 × 0.03 + 0.4 × 0.08 [1] = 0.018 + 0.032 = 0.05 [1] (c) Only the defective paths are possible, with probabilities 0.018 (machine A) and 0.032 (machine B) out of 0.05 in total [1] P(B | defective) = 0.032/0.05 [1] = 0.64 (or 16/25) [1] Examiner insight: In a “given” question, say which paths remain and divide by their combined probability.

10. (a) 0.6 × 0.6 × 0.6 = 0.216 [1] (b) One wrong answer in each of three positions, so 3 × 0.6 × 0.6 × 0.4 [1] = 0.432 [1] (c) 1 - P(none) = 1 - 0.4³ [1] = 1 - 0.064 = 0.936 [1] (d) P(qualifies) = 0.432 + 0.216 = 0.648 [1] Only these paths are possible, so P(all three | qualifies) = 0.216/0.648 [1] = 1/3 [1] Examiner insight: Counting the three orders for “exactly two” is a step that is easy to forget. Listing the three orders takes seconds.

11. (a) Heads 1/2 and tails 1/2 [1] From heads: red 4/10, white 6/10. From tails: red 5/8, white 3/8 [1] (b) 1/2 × 4/10 + 1/2 × 5/8 [1] = 0.2 + 0.3125 = 0.5125 = 41/80 [1] (c) P(heads and red) = 0.2 [1] P(from P | red) = 0.2/0.5125 [1] = 16/41 (about 0.390) [1] (d) P(heads) × P(red) = 0.5 × 0.5125 = 0.25625 [1] This is not equal to P(heads and red) = 0.2, so the events are not independent [1] Examiner insight: A written conclusion that links the two numbers to the word “independent” is better practice than two numbers left side by side.

Where marks are usually lost

• Adding the probabilities of events that can occur together.
• Multiplying for events that are linked, such as draws without replacement.
• Keeping the denominator at the original total after items are removed.
• Dividing by 90 or 120 when the question names a smaller group.
• Not stating a conclusion after an independence test.
• Forgetting the mixed paths when finding “exactly one” or “exactly two”.
• Leaving branches that do not sum to 1.

Next steps

• Condense the rules with the revision notes.
• Review the method in the study guide.
• Visit the course hub and tick the checklist.
• Use the free diagnostics to spot the next gap.
• Book a free trial class at /trial/.

Official syllabus

Source: the Cambridge IGCSE Statistics 0479 syllabus (examination year 2027), published by Cambridge University Press & Assessment. Covered here: Topic 7 Probability, outcomes 7.5 to 7.7. Hub: Cambridge IGCSE Statistics.