Ch 1 · 6.1 The Poisson Distribution · A Level Mathematics 9709
Topic 1 of 5 in Probability & Statistics 2 — A Level Mathematics 9709 — 55 lessons.
Introduction
Calls arriving at a help desk, typing errors on a page, cars passing a point in a minute: events that happen randomly at a steady average rate. The number of such events in a fixed interval follows a Poisson distribution. It needs just one number, the average rate, and it also gives a quick approximation to awkward binomial calculations. (Syllabus section 6.1, Paper 6.)
Key ideas — The Poisson formula
X sim Po(λ), where λ is the mean number of events in the interval:
🧑🏫 Teacher's explanation
Imagine you are standing by a busy road counting how many cars pass by every hour. You know that, on average, 10 cars pass every hour, but you don't know exactly how many will pass in the next specific hour. This randomness is what the Poisson distribution models. It is used for 'rare events' occurring in a fixed interval of time or space.
📖 Definitions
🔍 Worked example — Easy
A call center receives an average of 3 calls per minute. Calculate the probability that they receive exactly 2 calls in a given minute. Assume X sim Po(3).
🔍 Worked example — Standard
The number of typos in a page of a textbook follows a Poisson distribution with mean λ = 1.5. Find the probability that there are at least 2 typos on a page.
🔍 Worked example — Exam-style
A bacteria culture has a mean of 25 new bacteria every hour. Using the Poisson distribution, find the probability that there are more than 30 bacteria in an hour. (Use Normal approximation)
⚠️ Common errors
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🎯 Exam question
Key ideas — Mean and variance
E(X) = λ Var(X) = λ
🧑🏫 Teacher's explanation
Imagine you are standing by a busy road counting how many cars pass a specific lamp post every hour. You know that, on average, 10 cars pass per hour. This 'average' is your λ. Because the cars arrive randomly and independently, sometimes you might see 8 cars, sometimes 12, and occasionally 15. The Poisson distribution is the mathematical tool we use to predict the probability of seeing exactly k cars in that hour.
📖 Definitions
🔍 Worked example — Easy
A random variable X follows a Poisson distribution with parameter λ = 4. State the mean and the variance of X.
🔍 Worked example — Standard
In a Poisson distribution, the variance is 7.2. Find the mean of the distribution.
🔍 Worked example — Exam-style
A researcher observes that for a certain type of rare mutation, the number of occurrences follows a Poisson distribution. The mean number of mutations is 3. A sample of data shows a mean of 3.1 and a variance of 5.5. Comment on the suitability of using a Poisson model for this data.
⚠️ Common errors
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🎯 Exam question
(a) Find the value of λ. [1] (b) Calculate the probability that there are exactly 3 faults in a given period. [2] [3 marks]
Key ideas — When is Poisson a good model?
Events must occur
🧑🏫 Teacher's explanation
Imagine you are standing by a busy road counting how many cars pass you every hour. If cars arrive randomly, one doesn't 'cause' the next, and the average number of cars per hour stays roughly the same throughout the day, you are looking at a Poisson process. The Poisson distribution is the mathematical tool we use to predict the probability of seeing a specific number of these events in a fixed window of time or space.
📖 Definitions
🔍 Worked example — Easy
A call centre receives an average of 4 calls per hour. Find the probability that they receive exactly 3 calls in a given hour. Assume the number of calls follows a Poisson distribution.
🔍 Worked example — Exam-style
A radioactive source emits particles at a constant rate of 40 particles per second. Use a Normal approximation to find the probability that in a 1-second interval, the number of particles emitted is between 35 and 45 inclusive.
⚠️ Common errors
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🎯 Exam question
(a) Find the probability that the student receives exactly 20 emails in a day. [2] (b) Using a suitable approximation, find the probability that the student receives fewer than 30 emails in a day. [3] [5 marks]
Key ideas — Poisson approximation to the binomial
If X sim B(n, p) with n large and p small (roughly n > 50 and np < 5), then
🧑🏫 Teacher's explanation
Imagine you are monitoring a busy airport terminal. You want to know how many travelers pass through a specific gate in one hour. The number of trials (total travelers) is massive, but the probability of any single traveler choosing that specific gate is tiny. This is the perfect scenario for the Poisson distribution.
📖 Definitions
🔍 Worked example — Easy
A factory produces components where the probability of a component being defective is 0.02. In a batch of 100 components, find the probability that exactly 2 are defective using the Poisson approximation.
🔍 Worked example — Standard
A rare disease affects 0.5\% of a population. In a sample of 400 people, find the probability that at most 3 people have the disease using the Poisson approximation.
🔍 Worked example — Exam-style
A website receives an average of 4 clicks per minute. Using the Poisson distribution, find the probability that the website receives more than 2 clicks in a specific minute.
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🎯 Exam question
Key ideas — Normal approximation to the Poisson
If λ is large (roughly λ > 15), then X ≈ N(λ, λ). Use a continuity correction as for the binomial: P(X ≥ 45) ≈ P(Y > 44.5).
🧑🏫 Teacher's explanation
Imagine you are counting how many raindrops hit a specific tile on a sidewalk every minute. Because raindrops are tiny and random, the number of hits follows a Poisson distribution. When the average number of hits (λ) is very small, the distribution looks lopsided, like a slide. But as the average number of hits increases, the distribution becomes more symmetrical and bell-shaped.
📖 Definitions
🔍 Worked example — Standard
Given X sim Po(25), find P(X > 30) using a Normal approximation.
🔍 Worked example — Exam-style
A machine produces defects at a rate of λ = 40 per day. Find the probability that there are at least 45 defects in a day using the Normal approximation.
⚠️ Common errors
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🎯 Exam question
Worked examples
Emails arrive at an average of 3 per hour. Find the probability of exactly 2 in an hour, and of at least 3. X sim Po(3):
🔍 Worked example — Easy
Emails arrive at a server at an average rate of 3 per hour. Find the probability that exactly 2 emails arrive in a given hour, and the probability that at least 3 emails arrive.
🔍 Worked example — Exam-style
A shop sells an average of 40 newspapers a day. Using a normal approximation, find the probability that the shop sells at least 45 newspapers tomorrow.
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Practice
Answers: (1) 1 - e^-2.5(1 + 2.5) = 0.713. (2) Po(6): 1 - P(X ≤ 3) = 0.849. (3) Po(3): P(X ≤ 3) = 0.647. (4) N(50, 50): P(Y > 59.5) = P(Z > 1.343) = 0.0896. (5) accidents occur independently of one another, singly, and at a constant average rate over time.