10 free A Level Mathematics Probability & Statistics 2 questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.
This topic covers The Poisson distribution, Linear combinations of random variables, Continuous random variables, Sampling & estimation, Hypothesis tests. Work through the questions below, then generate a full Probability & Statistics 2 paper with a fresh set every time.
Generate a full Probability & Statistics 2 paper →The random variable X has probability density function f(x) = kx2 for 1 ≤ x ≤ 4, and 0 otherwise.
Show that k = 43.
Find E(X), giving your answer in exact form.
Find the median of X.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | k(1 − 14) = 1 ∫ kx−2 dx = 1 M1 correct result shown A1 | 2 |
| (b) | 43 ln 4 ∫ x × kx2 dx = k ln x M1 limits applied M1 43 ln 4 A1 | 3 |
| (c) | 85 ∫1m f(x) dx = ½ M1 k(1 − 1m) = ½ A1 85 A1 | 3 |
The number of accidents on a road averaged 3 per week. After a speed limit was introduced, there were 7 accidents in 5 weeks.
Test at the 5% significance level whether the mean number of accidents has decreased.
State what is meant by a Type I error in this context, and find the probability of making a Type I error in this test.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | P(X ≤ 7) = 0.0180 < 0.05: reject H0; evidence of a decrease H0: λ = 3 (per week), H1: λ < 3 B1 Po(15) used B1 P(X ≤ 7) attempted M1 0.0180 A1 comparison with 0.05 M1 conclusion in context, not definite A1 | 6 |
| (b) | concluding that the mean has decreased when it has not; critical region X ≤ 8, so P(Type I error) = 0.0374 correct description in context B1 critical region X ≤ 8 found M1 0.0374 A1 | 3 |
The numbers of emails received by Sam and by Jo in an hour have independent distributions Po(1.5) and Po(1.5).
Find the probability that together they receive more than 3 emails in an hour.
Given that they receive 4 emails in total in an hour, find the probability that Sam receives exactly 3 of them.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.3528 Po(3) B1 1 − P(X ≤ 3) M1 0.3528 A1 | 3 |
| (b) | 0.2500 P(S = 3)P(J = 1) ÷ P(total 4) M1 correct values or binomial with p = 0.5 A1 0.2500 A1 | 3 |
The masses of packets of rice are normally distributed with mean 414 g and standard deviation 24 g.
Find the probability that the mean mass of a random sample of 25 packets exceeds 417 g.
Find the smallest sample size for which the probability that the sample mean exceeds 417 g is less than 0.01.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.2660 X̄ ~ N(414, 57625) B1 standardise M1 0.2660 A1 | 3 |
| (b) | 347 z = 2.326 B1 324/√n > 2.326 M1 correct least n A1 | 3 |
The random variable X has probability density function f(x) = kx2 for 0 ≤ x ≤ 3, and 0 otherwise.
Find k.
Find E(X).
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 19 ∫ f(x) dx = 1 M1 19 A1 | 2 |
| (b) | 2.25 ∫ x f(x) dx M1 2.25 A1 | 2 |
A company claims that 20% of its customers are dissatisfied. In a random sample of 15 customers, 6 were dissatisfied.
Test at the 5% significance level whether the proportion is greater than claimed.
| Part | Answer and marking points | Marks |
|---|---|---|
| do not reject H₀ H₀: p = 0.2, H₁: p > 0.2 B1 B(15, 0.2) M1 P(X ≥ 6) = 0.06105 A1 compare with 0.05 M1 do not reject H₀: insufficient evidence A1 | 5 |
X ~ N(289, 52) and Y ~ N(105, 42) are independent.
Find P(X + 2Y > 508) and explain why this is different from P(X + Y1 + Y2 > 508), where Y1, Y2 are independent observations of Y.
| Part | Answer and marking points | Marks |
|---|---|---|
| 0.17 E = 499 B1 Var = 89 B1 standardise M1 0.17 A1 Var(X + Y1 + Y2) = 57 B1 so the probabilities differ B1 | 6 |
The number of calls to a helpline in a 5-minute period has the distribution Po(3.6).
Find the probability of exactly 2 calls in a 5-minute period.
Find the probability of at least 2 calls in a 10-minute period.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.177 e−3.6 × 3.62 ÷ 2! M1 0.177 A1 | 2 |
| (b) | 0.994 λ = 7.2 B1 1 − P(0) − P(1) M1 0.994 A1 | 3 |
Calls to a helpline arrive at random at an average rate of 2.5 per 10 minutes.
Find the probability of at least 2 calls in 10 minutes.
Find the probability of exactly 3 calls in 30 minutes.
Use a suitable approximation to find the probability of more than 50 calls in 3 hours.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.713 1 − P(0) − P(1) M1 0.713 A1 | 2 |
| (b) | 0.0389 λ = 7.5 M1 0.0389 A1 | 2 |
| (c) | normal approximation N(45, 45) B1 continuity correction 50.5 M1 standardise M1 probability A1 | 4 |
The mean time for a process is believed to be 53 minutes, with standard deviation 7.3 minutes. After changes, a random sample of 60 times has mean 53.34 minutes. Test at the 5% significance level whether the mean time has increased.
| Part | Answer and marking points | Marks |
|---|---|---|
| z = 0.361; do not reject H0: insufficient evidence of an increase H0: μ = 53, H1: μ > 53 B1 Standardise with σ/√n M1 z = 0.361 A1 Compare with 1.645 M1 Correct conclusion in context A1 | 5 |
In the 9709 syllabus, Probability & Statistics 2 covers The Poisson distribution, Linear combinations of random variables, Continuous random variables, Sampling & estimation, Hypothesis tests.
This page has 10 Probability & Statistics 2 questions with answers. For more, the free generator on mentorriyaz.in builds a new Probability & Statistics 2 paper with a full mark scheme each time you click.
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Meet Riyaz →Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in