Probability of Combined Events · Form 4
Probability of Combined Events: Practice Questions
Original SPM-style practice questions for Probability of Combined Events, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.
Original practice questions for Probability of Combined Events, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.
Multiple-choice (Paper 1 style)
Question 1
A fair coin is tossed once and a fair die numbered 1 to 6 is rolled once. How many possible outcomes are there in the sample space?
- A. 6
- B. 8
- C. 12
- D. 36
Question 2
A fair die numbered 1 to 6 is rolled twice. Find the probability that both rolls show a number greater than 4.
- A. 1/6
- B. 1/9
- C. 2/9
- D. 1/3
Question 3
A number is chosen at random from the integers 1 to 20. Find the probability that the number is divisible by 2 or by 5.
- A. 2/5
- B. 1/2
- C. 3/5
- D. 7/10
Question 4
A box contains 5 red marbles and 3 blue marbles. Two marbles are drawn at random one after another without replacement.
Find the probability that both marbles are red.
- A. 5/14
- B. 25/64
- C. 5/28
- D. 5/8
Question 5
The probability that it rains on a particular day is 0.35. Find the probability that it does not rain on that day.
- A. 0.35
- B. 0.55
- C. 0.75
- D. 0.65
Question 6
A student attempts two independent questions. The probability of solving the first is 0.6 and of solving the second is 0.5.
Find the probability that the student solves at least one question.
- A. 0.3
- B. 0.8
- C. 0.7
- D. 0.9
Question 7
A spinner lands on red with probability 1/4 on each spin. The spinner is spun twice.
Find the probability that it lands on red on both spins.
- A. 1/16
- B. 1/8
- C. 1/4
- D. 1/2
Question 8
Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.45. Find P(A ∪ B).
- A. 0.135
- B. 0.6
- C. 0.75
- D. 0.15
Question 9
Events A and B are independent with P(A) = 0.4 and P(B) = 0.7. Find P(A ∩ B).
- A. 1.1
- B. 0.3
- C. 0.11
- D. 0.28
Question 10
A bag contains 4 green sweets and 6 yellow sweets. One sweet is taken at random and eaten, then a second sweet is taken.
Find the probability that the first is green and the second is yellow.
- A. 6/25
- B. 4/15
- C. 2/5
- D. 24/81
Structured (Paper 2 style)
Question 1 (6 marks)
A box contains 3 red balls and 2 blue balls. A ball is drawn at random, its colour is noted, and it is returned to the box.
A second ball is then drawn. (a) Find the probability that both balls drawn are red.
(b) Find the probability that the two balls drawn are of different colours.
- Since each ball is returned before the next draw, the two draws are independent with P(red) = 3/5 and P(blue) = 2/5.
- (a) P(both red) = P(red) × P(red) = 3/5 × 3/5 = 9/25.
- (b) Different colours means (red, blue) or (blue, red): P = 3/5 × 2/5 + 2/5 × 3/5.
- P(different colours) = 6/25 + 6/25 = 12/25.
Question 2 (6 marks)
A basket contains 7 apples, of which 3 are rotten and 4 are good. Two apples are picked at random one after another without replacement.
(a) Find the probability that both apples are rotten. (b) Find the probability that exactly one of the two apples is rotten.
- Without replacement the second draw depends on the first; start with 7 apples (3 rotten, 4 good).
- (a) P(both rotten) = 3/7 × 2/6 = 6/42 = 1/7.
- (b) Exactly one rotten = (rotten, good) or (good, rotten) = 3/7 × 4/6 + 4/7 × 3/6.
- P(exactly one rotten) = 12/42 + 12/42 = 24/42 = 4/7.
Question 3 (5 marks)
In a class of 40 students, 25 study Physics, 18 study Chemistry, and 10 study both subjects. A student is chosen at random from the class.
(a) Find the probability that the student studies Physics or Chemistry. (b) Find the probability that the student studies neither subject.
- Use the addition rule: P(Physics or Chemistry) = P(Physics) + P(Chemistry) − P(both).
- (a) P(Physics or Chemistry) = 25/40 + 18/40 − 10/40 = 33/40.
- (b) 'Neither' is the complement of 'Physics or Chemistry'.
- P(neither) = 1 − 33/40 = 7/40.
Question 4 (7 marks)
Two archers, Amir and Bala, each shoot once at a target, independently of each other. The probability that Amir hits the target is 0.7 and the probability that Bala hits the target is 0.6.
(a) Find the probability that both archers hit the target. (b) Find the probability that exactly one archer hits the target.
(c) Find the probability that at least one archer hits the target.
- The shots are independent. P(Amir hits) = 0.7, P(Amir misses) = 0.3; P(Bala hits) = 0.6, P(Bala misses) = 0.4.
- (a) P(both hit) = 0.7 × 0.6 = 0.42.
- (b) Exactly one = (Amir hit, Bala miss) or (Amir miss, Bala hit) = 0.7 × 0.4 + 0.3 × 0.6.
- P(exactly one) = 0.28 + 0.18 = 0.46.
- (c) At least one = 1 − P(both miss) = 1 − 0.3 × 0.4 = 1 − 0.12 = 0.88.
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
What kinds of questions appear in the Paper 1 and Paper 2 sets for this topic?
Paper 1 items are multiple-choice, checking whether you can quickly identify the correct probability for combined events such as 'A or B' or 'A and B'. Paper 2 items are structured, often built around a tree diagram or table, requiring full working for independent or mutually exclusive events.
Why should I try to solve the problem myself before checking the answer?
Attempting the question first forces you to decide whether events are independent or mutually exclusive, which rule to apply, and how to read a tree diagram correctly. Checking the solution too soon hides these decision points, so you may not notice the same gap next time it appears.
What errors do students usually make with combined-event probability?
A frequent mistake is adding probabilities for events that are not mutually exclusive, or multiplying probabilities for events that are not independent. Students also mislabel branches on a tree diagram or forget that probabilities along one path must be multiplied while separate paths are added.