Probability & Statistics 2 (for Paper 6): past paper questions

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• Question 1May/June 2026, paper 61Structured5 marks

  The masses, in grams, of apples of a certain type have the distribution $\mathrm{N}(58.4,{6.3}^2)$. The apples are packed in bags, with each bag containing 10 randomly chosen apples.

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• Question 1Oct/Nov 2025, paper 61Structured9 marks

  The random variables $X$ and $Y$ have independent distributions $X\sim \mathrm{P}\mathrm{o}(3)$ and $Y\sim \mathrm{P}\mathrm{o}(2)$ respectively. Find $\mathrm{P}(2<X<5)$.

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• Question 1Oct/Nov 2024, paper 61Structured4 marks

  The heights of a certain species of deer are known to have standard deviation 0.35 m. A zoologist takes a random sample of 150 of these deer and finds that the mean height of the deer in the sample is 1.42 m. Calculate a 96% confidence interval for the population mean height.

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• Question 1Oct/Nov 2023, paper 61Structured3 marks

  A random variable $X$ has the distribution $\mathrm{N}(410,400)$. Find the probability that the mean of a random sample of 36 values of $X$ is less than 405.

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• Question 2May/June 2026, paper 61Structured5 marks

  In the past, $2\mathrm{\%}$ of talking dolls made by a certain firm have been faulty. Following a change in the manufacturing process, a quality control officer suspects that the percentage of faulty talking dolls has increased.

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• Question 2Oct/Nov 2025, paper 61Structured8 marks

  The mean mass of packets of Trueleaf tea is supposed to be 500 grams. An inspector wishes to test whether this value is correct. He weighs 60 randomly chosen packets and notes the mass, $x$ grams, of each packet.

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• Question 2Oct/Nov 2024, paper 61Structured5 marks

  The masses, in kilograms, of small and large bags of wheat have the independent distributions $\mathrm{N}(16.0,0.4)$ and $\mathrm{N}(51.0,0.9)$ respectively.

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• Question 2Oct/Nov 2023, paper 61Structured4 marks

  In a survey of 300 randomly chosen adults in Rickton, 134 said that they exercised regularly. This information was used to calculate an $\alpha \mathrm{\%}$ confidence interval for the proportion of adults in Rickton who exercise regularly.

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• Question 3May/June 2026, paper 61Structured11 marks

  In the work done by a typesetting department, incorrect characters appear randomly and independently at a constant average rate of 5 incorrect characters per 100 pages. A book that is typeset in the department contains 500 pages.

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• Question 3Oct/Nov 2025, paper 61Structured6 marks

  The data produced by a certain data entry firm always include a small number of incorrect characters that occur at random. The proportion of incorrect characters is denoted by $p$, and experience has shown that $p=0.0001$.

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• Question 3Oct/Nov 2024, paper 61Structured6 marks

  The times, $T$ minutes, taken by a random sample of 75 students to complete a test were noted.

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• Question 3Oct/Nov 2023, paper 61Structured10 marks

  A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution. Assume that the owner is correct.

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• Question 4May/June 2026, paper 61Structured11 marks

  The score on one spin of a 5-sided spinner is denoted by the random variable $X$ with probability distribution as shown in the table.

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• Question 4Oct/Nov 2025, paper 61Structured4 marks

  The masses of a certain species of animal are known to be normally distributed with standard deviation $\sigma \text{ kg}$.

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• Question 4Oct/Nov 2023, paper 61Structured8 marks

  The masses, in kilograms, of chemicals $A$ and $B$ produced per day by a factory are modelled by the independent random variables $X$ and $Y$ respectively, where $X\sim \mathrm{N}(10.3,5.76)$ and $Y\sim \mathrm{N}(11.4,9.61)$.

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• Question 5May/June 2026, paper 61Structured10 marks

  $X$ and $Y$ are independent random variables such that $X\sim \mathrm{P}\mathrm{o}(2.5)$ and $Y\sim \mathrm{P}\mathrm{o}(\lambda )$. Find $\mathrm{P}(1⩽X⩽3)$.

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• Question 5Oct/Nov 2025, paper 61Structured6 marks

  It is known that $20\mathrm{\%}$ of households in a certain country contain more than 4 people. Laxmi believes that, in her town, the percentage is lower than $20\mathrm{\%}$. She chooses a random sample of 40 households in her town and notes the number which contain more than 4 people.

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• Question 5Oct/Nov 2024, paper 61Structured6 marks

  The lengths, in centimetres, of worms of a certain kind are normally distributed with mean $\mu$ and standard deviation 2.3. An article in a magazine states that the value of $\mu$ is 12.7. A scientist wishes to test whether this value is correct.

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• Question 5Oct/Nov 2023, paper 61Structured5 marksHas a diagram

  In the past the number of enquiries per minute at a customer service desk has been modelled by a random variable with distribution $\mathrm{P}\mathrm{o}(0.31)$. Following a change in the position of the desk, it is expected that the mean number of enquiries per minute will increase.

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• Question 6Oct/Nov 2025, paper 61Structured10 marks

  The masses, in kilograms, of large and small bags of potatoes have the independent distributions $\mathrm{N}(2.5,0.05)$ and $\mathrm{N}(0.8,0.02)$ respectively.

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• Question 6Oct/Nov 2024, paper 61Structured9 marks

  The numbers of customers arriving at service desks $A$ and $B$ during a 10-minute period have the independent distributions $\mathrm{P}\mathrm{o}(1.8)$ and $\mathrm{P}\mathrm{o}(2.1)$ respectively.

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• Question 6Oct/Nov 2023, paper 61Structured8 marksHas a diagram

  A continuous random variable $X$ takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, $a$ and $b$, of $X$ are such that $\mathrm{P}(a<X<b)=p$ and $\mathrm{P}(b<X<3)=\frac{13}{10}p$, where $p$ is a positive constant.

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• Question 1May/June 2026, paper 62Structured3 marks

  Ten random values of a variable $X$ are given below. $$\begin{array}{llllllllll}8.1& 8.3& 9.5& 10.1& 7.9& 8.7& 9.1& 8.8& 9.4& 9.0\end{array}$$ Calculate unbiased estimates of the population mean and variance of $X$.

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• Question 7Oct/Nov 2024, paper 61Structured14 marks

  The number of accidents per year on a certain road has the distribution $\mathrm{P}\mathrm{o}(\lambda )$. In the past the value of $\lambda$ was 3.3. Recently, a new speed limit was imposed and the council wishes to test whether the value of $\lambda$ has decreased.

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• Question 7Oct/Nov 2023, paper 61Structured12 marksHas a diagram

  A biologist wishes to test whether the mean concentration $\mu$, in suitable units, of a certain pollutant in a river is below the permitted level of 0.5. She measures the concentration, $x$, of the pollutant at 50 randomly chosen locations in the river.

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• Question 2May/June 2026, paper 62Structured5 marks

  Each week, Birgitte takes three train journeys. The times, in minutes, for these journeys have the independent distributions $A\sim \mathrm{N}(28,15)$, $B\sim \mathrm{N}(23,8)$ and $C\sim \mathrm{N}(99,24)$. For a randomly chosen week, find $\mathrm{P}(C>2(A+B))$.

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• Question 1Oct/Nov 2025, paper 62Structured8 marks

  The number, $X$, of used computers donated to a charity has a constant average rate of 2.4 computers per week. State a necessary condition for $X$ to have a Poisson distribution.

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• Question 1Oct/Nov 2024, paper 62Structured3 marks

  A random variable $X$ has the distribution $\mathrm{B}(4500000,\frac{1}{1000000})$. Use a Poisson distribution to calculate an estimate of $\mathrm{P}(X⩾4)$.

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• Question 1Oct/Nov 2023, paper 62Structured6 marks

  A random variable $X$ has the distribution $\mathrm{P}\mathrm{o}(25)$. Use the normal approximation to the Poisson distribution to find $\mathrm{P}(X>30)$.

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• Question 3May/June 2026, paper 62Structured6 marks

  The number of vehicles, $X$, passing a certain junction between 1.00 pm and 1.01 pm is modelled by a Poisson distribution. In the past, the population mean of $X$ was 2.4. After a change is made to the junction, the value of $X$ is noted on a randomly chosen day.

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