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A-Level Mathematics Past Papers & Drilling (Paper 5): Permutations, Probability
TUTORZONE ORIGINAL DRILLING MOCK · CAMBRIDGE A-LEVEL MATHEMATICS (9709) PAPER 5
Written to the Cambridge 9709 syllabus (2026-2027): Paper 5 Probability & Statistics 1, 50 marks, 1 h 15 min, 20% of A Level (6-8 structured questions, compulsory). This is a Paper 5 drilling mock: 8 questions for 50 marks with answers and techniques.
What is in A-Level Mathematics (9709) Paper 5?
Direct answer: Paper 5 Probability & Statistics 1, 50 marks, 1 h 15 min, 20% of A Level (6-8 structured questions, compulsory).
Assessment at a glance (2027)
| Component | Content | Weighting | Duration |
|---|---|---|---|
| Paper 5 (Probability & Statistics 1) | 6-8 structured questions, 50 marks | 20% A-Level | 1 h 15 min |
| Paper 6 (Probability & Statistics 2) | 6-8 structured questions, 50 marks | 20% A-Level | 1 h 15 min |
A-Level Statistics drilling technique: 8 questions in 1 h 15 min. Use nPr/nCr; write ‘1 – opposite’ for probabilities; E(X) = sum x p(x).
How this paper was built
This is a TutorZone original drilling mock: the questions follow the Cambridge 9709 structure (Paper 5, 50 marks, 20%) and the answers are our own, not Cambridge past papers or marking references.
Every answer was recomputed programmatically.
Pacing
| Section | Marks | Suggested time | What to do first |
|---|---|---|---|
| Paper 5 (Probability & Statistics 1) | 20% A-Level | about 9 min per question | permutations, probability, variables |
| Paper 6 (Probability & Statistics 2) | 20% A-Level | about 9 min per question | Poisson, hypothesis tests |
nPr vs nCr?
Direct answer: nPr counts order (permutation), nCr ignores order (combination).
Paper 5: statistics drilling (8 questions, 50 marks)
nPr/nCr; ‘1 – opposite’ for probabilities.
Question 1 (6 marks)
How many ways can 3 of 5 distinct people be arranged in a line?
Question 2 (6 marks)
Two fair coins are tossed. Find the probability of at least one head.
Question 3 (7 marks)
A discrete variable X takes 1, 2, 3 each with probability 1/3. Find E(X).
Question 4 (7 marks)
Find the mean of the data 4, 6, 8, 10.
Question 5 (6 marks)
How many ways can 2 of 6 people be chosen for a team?
Question 6 (6 marks)
A bag has 3 red and 2 blue balls. Find the probability of drawing a red ball.
Question 7 (6 marks)
For X ~ B(2, 0.5), find P(X = 0).
Question 8 (6 marks)
A fair die is rolled. Find the probability of an even number.
How is the expectation calculated?
Direct answer: E(X) = sum x p(x).
Paper 5 answers
Question 1
5P3 = 5 x 4 x 3 = 60 (6).
Question 2
P = 1 – P(both tails) = 1 – (1/2)^2 = 3/4 (6).
Question 3
E(X) = 1/3 + 2/3 + 3/3 = 2 (7).
Question 4
Mean = (4+6+8+10) / 4 = 7 (7).
Question 5
6C2 = 6! / (2! x 4!) = 15 (6).
Question 6
P(red) = 3 / 5 = 3/5 (6).
Question 7
P(X=0) = (0.5)^2 = 1/4 (6).
Question 8
Even = 2, 4, 6; P = 3/6 = 1/2 (6).
Which topics does this paper drill?
Direct answer: permutations, probability, expectation and the binomial distribution.
Where these marks are usually lost
| Mistake | What it looks like | What it costs | Instead |
|---|---|---|---|
| Perm/combine mixed | Order as C not P | Use nPr when order matters | 5P3 = 60 |
| At-least missing 1- | Only opposite | At least one = 1 – opposite | 1 – 1/4 = 3/4 |
| Expectation missing | Probabilities only | E(X) = sum x p(x) | E(X) = 2 |
How to use this paper
First drill: the full 1 h 15 min on all 8 questions with no references.
Mark it: check nPr/nCr and probability formulas.
Second drill: redo three to seven days later, aiming to finish inside 1 hour.
What should I do after drilling this paper?
Direct answer: log questions with wrong permutations or probabilities; redo in three to seven days; and drill Paper 6 Probability & Statistics 2.
FAQ
Is this a Cambridge past paper?
No. The questions and answers are TutorZone originals, not Cambridge past papers or marking references.
What does A-Level Mathematics Paper 5 test?
Probability & Statistics 1: permutations, probability, discrete variables, normal distribution, 50 marks, 20% of A Level.
nPr vs nCr?
nPr counts order (permutation), nCr ignores order (combination).
Official references
| Resource | What to check |
|---|---|
| Cambridge A-Level Mathematics 9709 syllabus (2026-2027) | Confirm the Paper 1-6 marks and times |
| https://www.tutorzone.com.hk/en-alevel-mathematics-9709-paper-4-mock-2027/ | A-Level Mathematics Paper 4 Mechanics drilling mock |
Related: practice area · A-Level Mathematics Paper 4 drilling mock
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