The Poisson distribution: past paper questions

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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 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 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 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 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 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 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 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 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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• 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 4May/June 2026, paper 62Structured10 marks

  The random variables $D$ and $E$ have the independent distributions $D\sim \mathrm{P}\mathrm{o}(1.7)$ and $E\sim \mathrm{P}\mathrm{o}(2.1)$. Find $\mathrm{P}(D+E<4)$.

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• Question 4Oct/Nov 2025, paper 62Structured7 marks

  An inspector believes that 18% of cups made at a certain factory contain flaws. The factory owner claims that the true percentage is less than 18%. The inspector examines a random sample of 40 cups and finds that 3 of them contain flaws.

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• Question 5Oct/Nov 2024, paper 62Structured11 marks

  A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions $\mathrm{P}\mathrm{o}(3.7)$ and $\mathrm{P}\mathrm{o}(2.6)$ respectively. A bag of sweets is chosen at random.

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• Question 7Oct/Nov 2023, paper 62Structured9 marks

  A random variable $X$ has the distribution $\mathrm{P}\mathrm{o}(2.4)$. Find $\mathrm{P}(2⩽X<4)$.

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• Question 4May/June 2026, paper 63Structured11 marks

  Cars arrive at a junction at a constant average rate of 3.2 per minute. State a condition for the number of cars arriving per minute to have a Poisson distribution.

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• Question 1May/June 2025, paper 61Structured3 marks

  It is known that 1% of houses in a certain area have a wind turbine. A random sample of 400 houses in this area is chosen for a survey on domestic heating. The number of houses in the sample that have a wind turbine is denoted by $X$.

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• Question 1May/June 2024, paper 61Structured3 marks

  A bus station has exactly four entrances. In the morning the numbers of passengers arriving at these entrances during a 10-second period have the independent distributions $\mathrm{P}\mathrm{o}(0.4)$, $\mathrm{P}\mathrm{o}(0.1)$, $\mathrm{P}\mathrm{o}(0.2)$ and $\mathrm{P}\mathrm{o}(0.5)$.

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• Question 1May/June 2023, paper 61Structured6 marks

  In a certain country, 20 540 adults out of a population of 6 012 300 have a degree in medicine. Use an approximating distribution to calculate the probability that, in a random sample of 1000 adults in this country, there will be fewer than 4 adults who have a degree in medicine.

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• Question 5May/June 2025, paper 61Structured11 marks

  The random variables $W$ and $X$ have the independent distributions $\mathrm{P}\mathrm{o}(1.2)$ and $\mathrm{P}\mathrm{o}(2.3)$ respectively. Find $\mathrm{P}(3⩽W+X⩽5)$.

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• Question 5May/June 2024, paper 61Structured8 marks

  Sales of cell phones at a certain shop occur singly, randomly and independently. State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.

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• Question 7May/June 2023, paper 61Structured14 marks

  The number of accidents per week at a certain factory has a Poisson distribution. In the past the mean has been 1.9 accidents per week. Last year, the manager gave all his employees a new booklet on safety.

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• Question 3May/June 2025, paper 62Structured4 marks

  The random variable $X$ has the distribution $\text{Po}(15)$. Write down an expression in terms of $\text{e}$ for $\text{P}(X=12)$.

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• Question 7May/June 2024, paper 61Structured11 marks

  Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number $X$ of days on which she sees it.

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• Question 2May/June 2023, paper 62Structured5 marks

  The random variable $W$ has a Poisson distribution. State the relationship between $\mathrm{E}(W)$ and $\mathrm{Var}(W)$.

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• Question 6May/June 2025, paper 62Structured10 marks

  Use suitable approximating distributions to answer the following. The random variable $W$ has the distribution $\text{B}(700,0.005)$. Find $\text{P}(W⩾4)$.

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• Question 1May/June 2024, paper 62Structured5 marks

  A random variable $X$ has the distribution $\mathrm{P}\mathrm{o}(145)$. Use a suitable approximating distribution to calculate $\mathrm{P}(X⩽150)$.

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• Question 4May/June 2023, paper 62Structured10 marks

  The number, $X$, of books received at a charity shop has a constant mean of 5.1 per day. State, in context, one condition for $X$ to be modelled by a Poisson distribution.

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• Question 1May/June 2025, paper 63Structured8 marks

  At a certain shop, customers arrive independently and randomly at a constant average rate of 23.4 per hour. Find the probability that, in a randomly chosen 1-minute period, at least 2 customers arrive.

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