Further Probability & Statistics (for Paper 4): past paper questions

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• Question 1May/June 2026, paper 41Structured7 marks

  The manager of a village store is interested in the shopping habits of his customers. From previous records, the median number of items bought per customer was found to be 8. The manager believes the median number of items bought per customer has increased.

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• Question 1Oct/Nov 2025, paper 41Structured6 marks

  A group of 10 school children are asked to estimate the size of an angle ${\theta }^{\circ }$ in a given acute angled triangle.

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• Question 1Oct/Nov 2024, paper 41Structured6 marks

  Ellie is investigating the heights of two types of beech tree, $A$ and $B$, in a certain region.

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• Question 1Oct/Nov 2023, paper 41Structured6 marks

  Maya is an athlete who competes in 1500-metre races. Last summer her practice run times had mean 4.22 minutes. Over the winter she has done some intense training to try to improve her times.

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• Question 2May/June 2026, paper 41Structured10 marks

  Three different bus companies, $A$, $B$ and $C$, record the reliability of their service. Each bus journey is recorded as being on time, late or cancelled. A reliable bus service is considered to be one that is on time.

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• Question 2Oct/Nov 2025, paper 41Structured6 marks

  The manager of a car park claims that the number of cars entering the car park follows a Poisson distribution with mean 2.8. The numbers of cars entering the car park are recorded on a working day during successive 5-minute periods.

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• Question 3Oct/Nov 2023, paper 41Structured8 marks

  Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day.

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• Question 3May/June 2026, paper 41Structured6 marks

  Students studying geography at a college choose either human geography or physical geography. A teacher is investigating the times taken by students to complete assignments of equal difficulty in human geography and physical geography.

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• Question 3Oct/Nov 2025, paper 41Structured4 marks

  A random sample of 10 newborn baby boys is taken and their masses in kg are recorded. From this sample, the population standard deviation of all newborn baby boys is estimated as 0.6 kg.

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• Question 4Oct/Nov 2025, paper 41Structured5 marks

  A researcher believes that the median $m$ of a population has changed from its known previous value $m_0$. The researcher collects a random sample of size 28. She ranks the data and calculates a test statistic $T$ using the Wilcoxon signed-rank test.

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• Question 5Oct/Nov 2024, paper 41Structured9 marks

  Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is $\frac{1}{3}$ and the third coin is biased so that the probability of obtaining a head is $\frac{1}{5}$.

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• Question 5Oct/Nov 2023, paper 41Structured10 marks

  The random variable $X$ has the geometric distribution $\mathrm{G}\mathrm{e}\mathrm{o}(p)$. Show that the probability generating function of $X$ is $\frac{pt}{1-qt}$, where $q=1-p$.

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• Question 5May/June 2026, paper 41Structured8 marks

  Margaret has developed a new revision course to help students improve their ability in mathematics. The students each sit a mathematics test before attending the course and a test of similar difficulty after attending the course.

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• Question 6Oct/Nov 2024, paper 41Structured8 marks

  Ansal is investigating the wingspans of Monarch butterflies in two different regions, $X$ and $Y$. He takes a random sample of 8 Monarch butterflies from region $X$ and records their wingspans, $x\text{ cm}$. His results are as follows.

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• Question 6Oct/Nov 2023, paper 41Structured10 marks

  A school is conducting an experiment to see whether the distance that children can throw a ball increases in hot weather. On a cold day, all the children at the school were asked to throw a ball as far as possible. The distances thrown were measured and recorded.

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• Question 6May/June 2026, paper 41Structured10 marks

  The discrete random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ defined by $${\mathrm{G}}_X(t)=\frac{k}{t}(at+1{)}^2\mathrm{.}$$ It is given that $\mathrm{E}(X)=\frac{1}{3}$. Show that $a=2$ and find the value of $k$.

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• Question 6Oct/Nov 2025, paper 41Structured9 marks

  Nine athletes in a club have a new coach. The coach adopts a new training programme which he believes will reduce the race times of these athletes. Each athlete completes a 1500 m time trial before and after completing the new training programme.

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• Question 1Oct/Nov 2024, paper 42Structured4 marks

  A scientist is investigating the lengths of the leaves of a certain type of plant. The scientist assumes that the lengths of the leaves of this type of plant are normally distributed.

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• Question 1Oct/Nov 2023, paper 42Structured6 marks

  A factory produces small bottles of natural spring water. Two different machines, $X$ and $Y$, are used to fill empty bottles with the water. A quality control engineer checks the volumes of water in the bottles filled by each of the machines.

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• Question 7Oct/Nov 2025, paper 41Structured10 marks

  A discrete random variable $X$ takes values $r=0,1,2$ with probabilities $\mathrm{P}(X=r)$ as given in the following table.

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• Question 2Oct/Nov 2024, paper 42Structured8 marks

  The random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ given by $${\mathrm{G}}_X(t)=\frac{1}{5}+pt+q{t}^2,$$ where $p$ and $q$ are constants. Given that $\mathrm{E}(X)=1.1$, find the numerical value of $\mathrm{Var}(X)$.

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• Question 3Oct/Nov 2023, paper 42Structured10 marks

  Toby has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 3 marbles from the bag, without replacement. The random variable $X$ is the number of red marbles that Toby obtains.

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• Question 1Oct/Nov 2025, paper 42Structured6 marks

  A large company claims that the median salary of its employees is $32500.Thesalaries($) of 15 randomly selected employees are listed below.

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• Question 2Oct/Nov 2025, paper 42Structured7 marks

  A factory produces packets of biscuits. The total mass of biscuits in a packet has a normal distribution with mean $\mu$. A random sample of 12 packets is taken and the mass of the contents of each packet, $x\text{ g}$, is recorded.

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• Question 5Oct/Nov 2024, paper 42Structured9 marks

  Dev owns a small company which produces bottles of juice. He uses two machines, $X$ and $Y$, to fill empty bottles with juice. Dev is investigating the volumes of juice in the bottles.

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• Question 5Oct/Nov 2023, paper 42Structured16 marks

  A company is deciding which of two machines, $X$ and $Y$, can make a certain type of electrical component more quickly.

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• Question 3Oct/Nov 2025, paper 42Structured8 marks

  A traffic expert claims that the number of breakdowns occurring each day on a busy section of a motorway follows a Poisson distribution with mean 0.7. The number of breakdowns each day over a 200-day period was recorded.

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• Question 6Oct/Nov 2024, paper 42Structured9 marks

  A sports college keeps records of the times taken by students to run one lap of a running track. The population median time taken is 51.0 seconds.

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• Question 6May/June 2026, paper 42Structured10 marks

  The discrete random variable $X$ has probability generating function ${\mathrm{G}}_X(t)$ defined by $${\mathrm{G}}_X(t)=\frac{k}{t}(at+1{)}^2\mathrm{.}$$ It is given that $\mathrm{E}(X)=\frac{1}{3}$. Show that $a=2$ and find the value of $k$.

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• Question 5Oct/Nov 2025, paper 42Structured9 marks

  An engineer is comparing the tensile strengths of steel rods made from two machines, $A$ and $B$. The engineer randomly selects 8 rods from machine $A$ and 6 rods from machine $B$. The tensile strengths, in appropriate units, are given in the following table.

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