Probability practice problems GCSE Maths, Topic 5
Probability scales, expected outcomes, relative frequency, independent events, tree diagrams, Venn diagrams and conditional probability. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.
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Key equations for probability
| P(not A) = 1 − P(A) | complement |
|---|---|
| P(A and B) = P(A) × P(B) | independent events |
| expected number = probability × trials | expected outcomes |
A formulae sheet is provided in the 2025, 2026 and 2027 GCSE Maths exams. Answers within 2.5% are marked correct, and you can type fractions such as 23/20. Every topic, Foundation and Higher, is in our GCSE Maths topic list.
Warm-up questions
Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.
Hint
Probabilities of all outcomes add up to 1.
Worked solution
- 1 − 0.35 = 0.65.
Hint
Blue counters out of all counters.
Worked solution
- 5 blue out of 8 counters: 5/8 (0.625).
Hint
Expected number = probability × number of trials.
Worked solution
- (1/6) × 120 = 20: C.
Exam-style questions
Questions at the level of the GCSE Maths exam. The ones marked Challenge combine more than one idea.
Hint
Multiply for independent events.
Worked solution
- 0.3 × 0.6 = 0.18.
Hint
After the first red is taken, there are 2 red left out of 7.
Worked solution
- P = 3/8 × 2/7 = 6/56.
- = 3/28 (about 0.107).
Hint
Draw a Venn diagram, starting with the 5 in the overlap.
Worked solution
- Football only 13, tennis only 7, both 5: total 25.
- Neither = 30 − 25 = 5, so P = 5/30 = 1/6.
Hint
Relative frequency = frequency ÷ trials.
Worked solution
- 46 ÷ 200 = 0.23.
Hint
Given they play football, only the 18 football players count.
Worked solution
- 5 of the 18 football players also play tennis.
- P = 5/18 (about 0.278).
Exam question practice
One original exam-style question in the style of a GCSE paper. Answer it in full, with working, before you open the mark scheme: most marks are for method, so show every step.
Exam-style question style · 5 points · about 6 minutes
The probability that it rains on Monday is 0.3. If it rains, the probability that Ali is late is 0.4. If it does not rain, the probability that Ali is late is 0.1.
- (a) Draw a probability tree diagram for this information.
- (b) Work out the probability that Ali is late on Monday.
- (c) Ali is late on Monday. What is the probability that it rained? (Higher)
Write your full answer on paper first, then open the scoring guide and tick each point you earned.
Scoring guide and model answer
Part (a)
First branches: rain 0.3, no rain 0.7. Then late 0.4 and not late 0.6 after rain; late 0.1 and not late 0.9 after no rain.
Part (b)
0.3 × 0.4 + 0.7 × 0.1 = 0.12 + 0.07 = 0.19.
Part (c)
P(rain | late) = 0.12/0.19 = 12/19, about 0.632.
Your mark: 0 of 5
Common mistakes in probability
- Adding probabilities that should be multiplied, along tree diagram branches.
- Forgetting to reduce the total when picking without replacement.
- Counting the overlap twice in a Venn diagram.
- Using the whole group as the denominator in a conditional probability.
Keep going
Revise the ideas behind these questions:
Then try real exam questions: GCSE Maths past papers, with official scoring guidelines. · All GCSE Maths units