10 free IB Mathematics: Applications & Interpretation Statistics & Probability questions written in the style of IB Mathematics: Applications & Interpretation papers, each with its answer and mark scheme. 2 of them are A* challenge questions, built to catch the mistakes strong students still make.
This topic covers Descriptive statistics, Correlation & regression, Probability, Binomial & normal distributions, χ² tests, t-tests, Poisson distribution, Markov chains, Confidence intervals, Hypothesis testing & errors. Work through the questions below, then generate a full Statistics & Probability paper with a fresh set every time.
Generate a full Statistics & Probability paper →The table shows the scores of 32 students. The mean score is 2.75.
| Score | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Frequency | 6 | p | 8 | q | 4 |
Find the values of p and q.
| Part | Answer and marking points | Marks |
|---|---|---|
| p = 9, q = 5 p + q = 14 A1 Σfx: 6 + 2p + 24 + 4q + 20 = 88 A1 solving simultaneously (M1) p = 9 A1 q = 5 A1 | 5 |
For events A and B, P(A) = 0.4, P(B) = 0.4 and P(A | B) = 0.75.
Find P(A ∩ B).
Find P(A ∪ B).
Find P(B | A′).
Determine whether A and B are independent, giving a reason.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.300 P(A | B) × P(B) (M1) 0.300 A1 | 2 |
| (b) | 0.500 P(A) + P(B) − P(A ∩ B) (M1) 0.500 A1 | 2 |
| (c) | 0.167 P(B ∩ A′)P(A′) = 0.1000.600 (M1) 0.167 A1 | 2 |
| (d) | not independent P(A | B) = 0.75 ≠ P(A) = 0.4, so not independent R1 | 1 |
Each year 20% of the customers of company X move to company Y, and 10% of the customers of company Y move to company X. Initially X has 2000 customers and Y has 2500.
Write down a transition matrix T for the movement of customers.
Find the eigenvalues and corresponding eigenvectors of T.
Hence write down matrices P and D such that T = PDP−1.
Find an expression for the number of customers of X after n years.
Write down the number of customers X can expect in the long term.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.8 0.10.2 0.9 columns sum to 1 (M1) 0.8 0.10.2 0.9 A1 | 2 |
| (b) | λ = 1, λ = 0.7 det(T − λI) = 0 (M1) λ = 1, λ = 0.7 A1 eigenvector for 1: 0.10.2 (or a multiple) A1 eigenvector for 0.7: 1−1 A1 | 4 |
| (c) | P, D P = 0.1 10.2 −1 A1 D = 1 00 0.7 A1 | 2 |
| (d) | 1500 + 500(0.7)n use of Tn = PDnP−1 M1 correct P−1 A1 xn = 1500 + 500(0.7)n A2 form A + B(λ)n A1 | 5 |
| (e) | 1500 1500 A1 | 1 |
Two judges award scores to 6 contestants.
| Contestant | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| Judge 1 | 61 | 50 | 65 | 70 | 58 | 45 |
| Judge 2 | 68 | 56 | 72 | 75 | 65 | 61 |
Find Spearman’s rank correlation coefficient, rs.
Comment on the level of agreement between the judges.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.943 ranking both sets of scores (M1) correct ranks A1 Pearson’s coefficient applied to the ranks (M1) rs = 0.943 A1 | 4 |
| (b) | strong agreement strong positive agreement R1 | 1 |
The probability that a seed germinates is 0.2. A packet contains 20 seeds.
Find the probability that exactly 4 seeds germinate.
Find the probability that more than 4 seeds germinate.
Find the expected number and the variance of the number of seeds that germinate.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.218 B(20, 0.2) (M1) 0.218 A1 | 2 |
| (b) | 0.370 1 − P(X ≤ 4) (M1) 0.370 A1 | 2 |
| (c) | 4, 3.2 E = 4 A1 Var = 3.2 A1 | 2 |
The volume of drink in bottles has mean 1000 ml and standard deviation 15 ml. The distribution of the volumes is not known.
A sample of 40 bottles is taken. State the approximate distribution of the sample mean X̄, giving a reason.
Find the probability that the sample mean is less than 999 ml.
A different sample of 40 bottles has mean 996 ml. Find a 95% confidence interval for the population mean, assuming σ = 15.
Hence comment on the claim that the mean volume is 1000 ml.
Find the least sample size needed for the width of a 95% confidence interval to be at most 4 ml.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | X̄ ~ N(1000, 5.63) central limit theorem, n large R1 mean 1000 A1 variance 15240 = 5.63 A1 | 3 |
| (b) | 0.337 normal CDF with standard error (M1) 0.337 A1 | 2 |
| (c) | 991.4 < μ < 1001 z-interval (M1) 996 ± 1.96 × 15√40 (A1) 991.4 < μ < 1001 A1 | 3 |
| (d) | supported 1000 is in the interval R1 the claim is supported A1 | 2 |
| (e) | 217 2 × 1.96 × 15√n ≤ 4 (M1) solving for n (M1) 217 A1 | 3 |
The number of books read by a group of students last month is shown.
| Books | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | 9 | 14 | 4 | 4 | 10 | 7 |
Find the mean and the standard deviation of the number of books read.
Find the median.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | mean 2.27, sd 1.79 GDC with frequencies (M1) mean 2.27 A1 sd 1.79 A1 | 3 |
| (b) | 2 48 values, median position 24.5 (M1) 2 A1 | 2 |
In a class of 54 students, 21 study French (F), 16 study Spanish (S) and 10 study both.
Find the number of students who study neither.
Find P(F ∪ S) and P(S | F).
Determine whether F and S are independent.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 27 27 A1 | 1 |
| (b) | 2754, 1021 2754 A1 conditional on F (M1) 1021 A1 | 3 |
| (c) | not independent P(F)P(S) = 0.115, P(F ∩ S) = 0.185 M1 not equal, so not independent A1 | 2 |
The lengths of a species of fish are normally distributed with mean 65 mm and standard deviation 12 mm.
Find the probability that a fish is longer than 77 mm.
8 fish are caught. Find the probability that exactly 2 are longer than 77 mm.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.159 normal CDF (M1) 0.159 A1 | 2 |
| (b) | 0.250 X ~ B(8, 0.159) (M1)(A1) 0.250 A1 | 3 |
The masses of apples are normally distributed with mean 50 g and standard deviation 8 g.
Find the probability that an apple has mass between 42 g and 55 g.
Find the mass exceeded by 10% of apples.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 0.575 GDC normalcdf (M1) 0.575 A1 | 2 |
| (b) | 60.3 g inverse normal (M1) 60.3 A1 | 2 |
In the SL & HL syllabus, Statistics & Probability covers Descriptive statistics, Correlation & regression, Probability, Binomial & normal distributions, χ² tests, t-tests, Poisson distribution, Markov chains, Confidence intervals, Hypothesis testing & errors.
This page has 10 Statistics & Probability questions with answers. For more, the free generator on mentorriyaz.in builds a new Statistics & Probability paper with a full mark scheme each time you click.
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Meet Riyaz →Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in