4.17 The Poisson distribution questions for IB Maths AI

44 questions on The Poisson distribution, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 4 Statistics and probability
HL only

Questions by paper and level

Paper	Questions
Paper 1	20
Paper 2	23
Paper 3	1
Every paper	44

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00961Paper 1 · 5 marks

The number of emails that Dr Okafor receives in one hour of her working day follows a Poisson distribution with mean 6.4.

She records the number of emails received in each of 40 randomly chosen working hours. Let $\overset{ˉ}{X}$ be the mean number of emails per hour for these 40 hours.

(a)
Explain why the central limit theorem can be used to approximate the distribution of $\overset{ˉ}{X}$ , even though the number of emails in one hour is not normally distributed.
1 mark
(b)
Write down the approximate distribution of $\overset{ˉ}{X}$ , stating its parameters.
2 marks
(c)
Find the probability that $\overset{ˉ}{X}$ is greater than 7.
2 marks

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Question 2 · AI-00759Paper 2 · 13 marks

Seismologists record, for one region, the mean number of earthquakes per year, $N$ , with magnitude at least $M$ .

$M$	3.0	3.5	4.0	4.5	5.0	5.5	6.0
$N$	22.4	7.9	3.1	1.02	0.37	0.12	0.041

The Gutenberg–Richter law states that $\log_{10} N = a - b M$ , where $a$ and $b$ are constants.

(a)
Find the equation of the regression line of $\log_{10} N$ on $M$ , and hence write down the values of $a$ and $b$ .
3 marks
(b)
Show that the law can be written in the form $N = \frac{c}{10^{b M}}$ , and find the value of $c$ .
2 marks
(c)
Find the mean number of earthquakes per year with magnitude at least 7.0. Give your answer as a decimal.
2 marks
(d)
Hence estimate how often, on average, an earthquake of magnitude at least 7.0 occurs in this region.
1 mark
(e)
The number of earthquakes of magnitude at least 7.0 in this region in a period of 50 years is modelled by a Poisson distribution. Find the probability that at least one such earthquake occurs in the next 50 years.
3 marks
(f)
State two assumptions made in using a Poisson distribution in part (e).
2 marks

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Question 3 · AI-01642Paper 3 · 27 marks

This question investigates whether the opening of a new marina has increased the number of boats arriving at a canal lock.

Boats on the Ellesmere Canal must pass through Hartwell Lock. The lock keeper, Ingrid, records the number of boats arriving at the lock in 40 randomly chosen one-hour periods during the summer.

Number of boats in one hour	0	1	2	3	4	5	6	7
Frequency	2	4	9	9	7	5	3	1

(a)
Ingrid wants to model the number of boats arriving in one hour by a Poisson distribution.
State two conditions that must hold for this model to be appropriate.
2 marks
(b)(i)
Calculate the mean and the variance of the number of boats arriving in one hour, according to Ingrid's data.
2 marks
(b)(ii)
Explain how your answers to part (b)(i) support the use of a Poisson model.
1 mark
(c)(i)
Ingrid models the number of boats arriving in one hour by a Poisson distribution with mean 3.2.
Find the probability that exactly 5 boats arrive in a one-hour period.
2 marks
(c)(ii)
Given that at least 3 boats arrive in a two-hour period, find the probability that fewer than 6 boats arrive in that period.
3 marks
(d)(i)
The lock operates in 30-minute cycles and can take at most 4 boats in each cycle. Any extra boats must wait for the next cycle.
Find the probability that more than 4 boats arrive during one 30-minute cycle.
2 marks
(d)(ii)
On one day the lock operates 20 cycles. The numbers of boats arriving in different cycles are independent.
Find the probability that more than 4 boats arrive in at least two of the cycles on that day.
3 marks
(e)(i)
The following summer a new marina opens upstream. Ingrid believes that the mean number of boats arriving per hour has increased. She will count the number of boats, $T$ , arriving during one randomly chosen five-hour period and test her belief at the 5 % significance level.
State the null and alternative hypotheses for this test.
1 mark
(e)(ii)
Find the critical region for this test.
3 marks
(e)(iii)
During the five-hour period, 25 boats arrive at the lock.
State the conclusion of the test, in context. Give a reason for your answer.
2 marks
(f)
Find the probability of a Type I error for this test.
2 marks
(g)
In fact, after the marina opens, the mean number of boats arriving is 4.4 per hour.
Find the probability of a Type II error for this test.
3 marks
(h)
Suggest how Ingrid could reduce the probability of a Type II error without increasing the significance level of the test.
1 mark

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Question 4 · AI-01059Paper 1 · 7 marks

The number of calls received by a bank's customer helpline in a 10-minute period can be modelled by a Poisson distribution with mean 4.5.

(a)
State two assumptions that must be made for the Poisson model to be appropriate, in the context of the question.
2 marks
(b)
Find the probability that exactly 6 calls are received in a 10-minute period.
2 marks
(c)
Find the probability that more than 30 calls are received in one hour.
3 marks

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Question 5 · AI-01088Paper 2 · 15 marks

A café in Lyon sells croissants and muffins. The number of croissants sold in an hour, $C$ , follows a Poisson distribution with mean $6.2$ . The number of muffins sold in an hour, $M$ , follows a Poisson distribution with mean $3.8$ . $C$ and $M$ are independent, and sales in different hours are independent.

(a)
Find the probability that exactly 5 croissants are sold in a given hour.
2 marks
(b)
The café makes a profit of 1.40 euros (EUR) on each croissant and 0.95 EUR on each muffin. The cost of running the pastry counter is 8 EUR per hour. The profit, $P$ EUR, from the pastry counter in one hour is therefore
$P = 1.40 C + 0.95 M - 8.$
(i) Find $E (P)$ .
(ii) Find $V a r (P)$ .
5 marks
(c)
State one reason why $P$ does not follow a Poisson distribution.
1 mark
(d)
Find the probability that at least 35 pastries in total are sold in a 3-hour period.
3 marks
(e)
The café is open for 8 hours each day. Let $Q$ be the total profit from the pastry counter in one day.
Use the central limit theorem to estimate the probability that $Q$ is less than 20 EUR.
4 marks

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Question 6 · AI-01060Paper 1 · 5 marks

Before publication, Kwame checks the manuscript of a novel for typing errors. The number of typing errors on a page of the manuscript follows a Poisson distribution with a mean of 0.8 errors per page.

(a)
Find the probability that a randomly chosen page has no typing errors.
2 marks
(b)
One chapter of the novel is 5 pages long. Find the probability that this chapter contains at least 3 typing errors.
3 marks

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