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Poisson & Binomial Distributions

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Edexcel A Level Further Maths · Further Statistics 1

Poisson & Binomial Distributions

Big idea: the Poisson distribution counts how many random, independent "things" happen in a fixed stretch of time or space — and when a Binomial distribution has a huge number of trials but a tiny chance of success each time, Poisson can step in and approximate it.

Quick Summary

The Poisson distribution, Po(λ), models the number of random events happening in a fixed interval of time or space.
Three conditions must hold: events happen independently, singly (never simultaneously), and at a constant average rate.
For X ~ Po(λ): mean = variance = λ. This is the Poisson's signature feature.
Scaling the interval scales λ by the same factor (e.g. double the time → double λ).
Independent Poisson variables add together: if X ~ Po(λ) and Y ~ Po(μ), then X + Y ~ Po(λ + μ) — but only over the same interval.
A Binomial B(n, p) can be approximated by Po(np) when n is large (n > 30) and p is small (np ≤ 10).

Topic 1

What Is the Poisson Distribution?

Imagine you're standing by a busy road counting how many buses pass in an hour, or counting how many typos are scattered across a page of text. In both cases you're not asking "will it happen or not" (that's a yes/no world — Binomial territory). You're asking "how many times will this happen?" when the event could, in theory, happen any number of times — 0, 1, 2, 3... all the way up.

That's exactly what the Poisson distribution is built for: counting random occurrences within a fixed window — a window of time (calls to a call centre per hour) or a window of space (flowers per square metre of a field).

Notation

X ~ Po(λ)

X is the number of times the event occurs in the interval. λ (lambda) is the Poisson parameter — and remarkably, it's both the mean and the variance of the distribution.

The Three Conditions

For a Poisson model to be valid, all three of these need to be roughly true:

Independence — one occurrence doesn't make another more or less likely. One customer walking into a shop shouldn't affect whether the next one does.
Singly — events can't happen at exactly the same instant (in time) or exactly the same point (in space). Two things can't occur "simultaneously."
Constant average rate — the rate of events doesn't change across the interval. Rush hour vs. 3am traffic would violate this if your interval spans both.

Think of it like this Picture raindrops landing on a pavement slab. Each drop lands independently of the others, no two drops can land in the exact same spot at the exact same moment, and (assuming the rain isn't getting heavier or lighter) the average rate of drops per minute stays steady. That's a textbook Poisson scenario.

When the Conditions Break Down

Examiners love asking you to criticise a Poisson model — meaning, explain why the real-world situation might not actually satisfy those three conditions. Classic example: counting people entering a restaurant.

Not independent — people often arrive because a friend invited them, so arrivals are linked.
Not singly — families or groups walk in together, arriving at the "same" moment.
Not constant rate — dinner time sees way more arrivals than 3pm on a quiet afternoon.

Even when a model isn't perfect, you can still say: "In order to proceed, we assume the occurrences are independent, happen singly, and occur at a constant average rate." That's the phrase examiners want to see when you're asked to state an assumption.

Examiner tip Never just say "the events aren't independent." Always tie it back to the actual context — "people entering may not be independent, because they could be invited in by others they know." Generic answers lose marks; contextualised ones score.

Practice Question

A shopkeeper wants to model the number of customers entering her small shop between 9am and 10am using a Poisson distribution. State two conditions required for this model to be appropriate, giving your answer in context.

Topic 2

Calculating Poisson Probabilities

If X ~ Po(λ), there's a formula that tells you the exact probability that X takes any particular value.

Poisson Probability Function

P(X = x) = e^−λ · λ^x / x! for x = 0, 1, 2, 3, …

In plain words: raise λ to the power of the value you want, divide by that value's factorial, and multiply by e to the power of minus λ. In practice, you'll almost always let your calculator's stats functions do this for you — but knowing the formula helps you understand why the numbers behave the way they do.

Notice something interesting: there's no upper limit on x. X could theoretically be 0, or 100, or a million — the probabilities just get closer and closer to zero as x grows huge. This works out neatly because of a bit of calculus magic: the Maclaurin series for e^λ guarantees that if you add up P(X = 0) + P(X = 1) + P(X = 2) + ... forever, you get exactly 1, just like every proper probability distribution must.

Rescaling the Interval

Here's a genuinely useful trick. Suppose you're told texts arrive at a rate of λ per hour. What if the question asks about a 3-hour window, or just 15 minutes? You don't need a new distribution from scratch — just scale λ by the same factor as the interval.

Rescaling Rule

Texts in 1 hour: Po(λ) → Texts in 3 hours: Po(3λ) → Texts in 15 min: Po(λ/4)

This only works because of the "constant average rate" condition — that's precisely what guarantees the rate scales linearly like this.

Cumulative Probabilities

Most calculators handle P(a ≤ X ≤ b) directly, but some only give you P(X ≤ b). If yours is the second type, or if a question mixes strict and non-strict inequalities, these identities are your best friend — because Poisson only takes whole number values, you can always convert a strict inequality into a weak one:

You want	Rewrite as	Why
P(X < x)	P(X ≤ x − 1)	e.g. X < 5 means 0,1,2,3,4, so this equals P(X ≤ 4)
P(X > x)	1 − P(X ≤ x)	"more than x" is everything above x
P(X ≥ x)	1 − P(X ≤ x − 1)	e.g. X ≥ 4 means everything except 0,1,2,3
P(a ≤ X ≤ b)	P(X ≤ b) − P(X ≤ a − 1)	cumulative up to b, minus everything below a

Common mistake

Mixing up which integer belongs in the inequality. If in doubt, physically list out the integers involved (e.g. "X ≥ 4 means 4, 5, 6, 7, …") before converting — this stops silly off-by-one errors that cost easy marks.

Worked Example

Rachel's Route to School

Pieces of rubbish occur at a rate of 2.5 per 100 metres. Assume a Poisson model is appropriate.

a) Exactly 3 pieces in 100 metres:
Let X = number of pieces of rubbish, so X ~ Po(2.5).
P(X = 3) = e^−2.5 × 2.5³/3! = 0.213763… ≈ 0.214

b) At least 1 piece in 50 metres:
Rescale the mean: 2.5 × (50/100) = 1.25, so X ~ Po(1.25).
P(X ≥ 1) = 1 − P(X = 0) = 0.713495… ≈ 0.713

c) No more than 10 pieces in 400 metres:
Rescale: 2.5 × (400/100) = 10, so X ~ Po(10).
P(X ≤ 10) = 0.583039… ≈ 0.583

d) At least 15 but fewer than 30 pieces in 1 km:
Rescale: 2.5 × (1000/100) = 25, so X ~ Po(25).
P(15 ≤ X < 30) = P(15 ≤ X ≤ 29) = 0.805494… ≈ 0.805

Practice Question

Flaws in a roll of fabric occur randomly at a rate of 1.6 per 5 metres. Find the probability that a 15-metre length of fabric contains fewer than 3 flaws.

Topic 3

Poisson Mean & Variance

This is arguably the single most testable fact about the Poisson distribution, so let it sink in properly:

The Defining Property

If X ~ Po(λ), then E(X) = λ and Var(X) = λ

Mean and variance are exactly the same number. No other common distribution has this quirk — it's the fingerprint of "Poisson-ness."

Using This to Test a Model

Because this property is so distinctive, exam questions often hand you raw sample data and ask: "is a Poisson model appropriate here?" Your job is to calculate the sample mean and sample variance and compare them.

If mean ≈ variance → this supports using a Poisson model (though you should still sanity-check the three conditions from Topic 1).
If mean and variance are clearly different → a Poisson model is not appropriate.

Sample Mean & Variance (from summary stats)

x̄ = Σx / n σ² = Σx²/n − x̄²

These are the standard formulas you'll already know from S1 — nothing new here, just applying them for a specific purpose.

Worked Example

Pineapple Pizzas

A student counts pieces of pineapple, x, on 50 pizzas. Σx = 1259, Σx² = 32964.

a) Mean and variance:
x̄ = 1259/50 = 25.18
σ² = 32964/50 − 25.18² = 659.28 − 634.0324 = 25.2476

b) Is Poisson appropriate?
To 1 d.p., both mean and variance equal 25.2. Because they're approximately equal, this suggests a Poisson distribution may be a suitable model for pieces of pineapple per pizza.

Examiner tip "Approximately equal" answers almost always need you to actually state the two numbers side-by-side (e.g. "mean = 25.2, variance = 25.2, so...") — don't just say "they're close." Show the comparison explicitly to bank the mark.

Practice Question

A researcher records the number of birdsong bursts heard in 40 one-minute intervals: Σx = 168, Σx² = 806. Calculate the sample mean and variance, and comment on whether a Poisson model would be appropriate.

Topic 4

Sum of Poisson Distributions

Here's a beautifully clean result. If two independent things are each Poisson, their sum is also Poisson — with the two means simply added together.

Addition Rule

If X ~ Po(λ) and Y ~ Po(μ) are independent, then X + Y ~ Po(λ + μ)

This extends to any number of independent Poisson variables: X₁ + X₂ + ... + Xₙ ~ Po(λ₁ + λ₂ + ... + λₙ).

Important condition To add Poisson variables together, they must all be modelling events over the same interval (e.g. all "per hour," or all "per 5 square metres"). They do NOT need to be the same type of event — pedestrians and cyclists can be added together as long as both rates are expressed over the same hour, say.

Common mistake

Students often try to "add" Poisson variables even when a question actually wants the probability of two separate events both happening — that's a multiplication of two individual probabilities, not a sum-of-Poisson-variables problem. See part (c) of the worked example below for exactly this trap.

Worked Example

Jovan's Window

Pedestrians pass at 5.2/hour. Cyclists pass at 3.8 per 30 minutes.

a) Exactly 15 pedestrians + cyclists total in 1 hour:
X = pedestrians, X ~ Po(5.2). Y = cyclists — convert to per hour: 3.8 × 2 = 7.6, so Y ~ Po(7.6).
X + Y ~ Po(5.2 + 7.6) = Po(12.8)
P(X + Y = 15) = 0.085640… ≈ 0.0856

b) At least 6 total in 15 minutes:
Rescale both to 15 minutes: X ~ Po(5.2/4) = Po(1.3). Y ~ Po(3.8/2) = Po(1.9).
X + Y ~ Po(1.3 + 1.9) = Po(3.2)
P(X + Y ≥ 6) = 0.105408… ≈ 0.105

c) At least 3 pedestrians AND at least 3 cyclists in 15 minutes:
Careful — this is NOT a sum! This is the probability of two independent events both occurring. X ~ Po(1.3), Y ~ Po(1.9), same as part (b).
P(X ≥ 3) = 0.142887…, P(Y ≥ 3) = 0.296279…
P(X ≥ 3) × P(Y ≥ 3) = 0.042334… ≈ 0.0423

d) Assumption made:
It has been assumed that the number of pedestrians and the number of cyclists are independent.

Practice Question

Emails arrive in Priya's inbox at a rate of 2.4 per 20 minutes. Text messages arrive at a rate of 1.5 per 10 minutes. Assuming independence, find the probability that Priya receives exactly 10 emails and texts combined in a 30-minute period.

Topic 5

Poisson Approximations of Binomials

The Binomial distribution B(n, p) counts successes in a fixed number of independent yes/no trials. But when n gets huge and p gets tiny — think "1 in 1000 people subscribe, out of 3000 website visitors" — calculating exact Binomial probabilities by hand becomes painful. This is exactly the situation the Poisson distribution was born to approximate.

The Approximation

X ~ B(n, p) ≈ X_p ~ Po(λ) where λ = np

Use this when n is large (a good rule of thumb: n > 30) and p is small, such that np ≤ 10. There's no hard boundary in maths — it's a judgment call the question will usually cue you into.

Why does this work? Because the Binomial's mean is np and its variance is np(1−p). When p is tiny, (1−p) is almost exactly 1, so the variance np(1−p) collapses to being almost equal to the mean np — which is precisely the Poisson's signature "mean = variance" property. In fact, the Poisson distribution can be derived mathematically as the limit of the Binomial as n → ∞ and p → 0 while np stays fixed.

When p is close to 1 instead If n is large but p is near 1 (not near 0), don't panic — just flip your focus to failures instead of successes. Let X′ = number of failures, so X′ ~ B(n, 1 − p). Now (1 − p) is small, and you can apply the Poisson approximation to X′.

Worked Example

Revision Website Subscribers

1 in 1000 people who check a revision website subscribe. The site got 3000 hits yesterday. Find the probability that more than 5 people subscribed.

Step 1 — Define the exact distribution:
Let X = number of subscribers. X ~ B(3000, 1/1000).

Step 2 — Check the approximation is valid:
n = 3000 is large. p = 1/1000 is small. ✓

Step 3 — Define the approximating distribution:
λ = np = 3000 × 1/1000 = 3, so X_p ~ Po(3).

Step 4 — Calculate:
P(X > 5) ≈ P(X_p > 5) = 1 − P(X_p ≤ 5) = 1 − 0.9161 = 0.0839 (3 s.f.)

Examiner tip Exam questions will almost always explicitly tell you to "use a Poisson approximation" — but you should still show the checking step (n large, p small) as part of your method, since it often carries its own mark.

Practice Question

A factory produces 5000 components a day. The probability that any individual component is defective is 0.0006. Use a suitable approximation to find the probability that fewer than 2 components are defective on a given day.

What to Memorise

Item	Statement
Notation	X ~ Po(λ), where λ > 0
Probability function	P(X = x) = e^−λλ^x/x!, for x = 0, 1, 2, …
Mean	E(X) = λ
Variance	Var(X) = λ (same as the mean!)
3 conditions	Independent · Singly · Constant average rate
Rescaling	Scale λ by the same factor as the interval
Sum rule	X ~ Po(λ), Y ~ Po(μ) independent ⟹ X+Y ~ Po(λ+μ)
Sum rule condition	Must be over the same interval; events need not be identical
Binomial → Poisson	B(n,p) ≈ Po(np) when n large (n>30), p small (np≤10)
p near 1 fix	Model failures instead: X′ ~ B(n, 1−p)
P(X<x)	= P(X ≤ x−1)
P(X>x)	= 1 − P(X ≤ x) = P(X ≥ x+1)

Concepts Checklist

I can state the three conditions required for a Poisson model, in context.
I can explain why a real-world scenario might fail to satisfy the Poisson conditions.
I know and can use the Poisson probability formula P(X=x) = e⁻λλˣ/x!.
I can find individual and cumulative Poisson probabilities on my calculator.
I can rescale λ when the interval of interest changes.
I can convert strict inequalities (<, >) into weak inequalities (≤, ≥) correctly.
I know that for Po(λ), mean = variance = λ.
I can use sample mean and variance to judge whether a Poisson model is appropriate.
I know that independent Poisson variables sum to another Poisson variable (same interval only).
I can tell the difference between "sum of two Poisson variables" and "probability of two independent events both happening."
I know when and how to approximate a Binomial with a Poisson distribution (large n, small p).
I can handle the case where p is close to 1 by modelling failures instead.

Exam Tips & Common Traps

Forgetting to rescale λ

If the question changes the interval (e.g. from "per hour" to "per 20 minutes"), you MUST rescale λ before doing any probability calculation. This is one of the most common lost-mark moments — always check the interval in the question matches the interval your λ was defined over.

Confusing "sum of Poissons" with "both events happening"

"Total number of X and Y" → add the Poisson variables. "At least 3 of X AND at least 3 of Y" → these are two separate probabilities that get multiplied, not added as random variables. Read the question phrasing very carefully.

Vague model criticism

Never write "the events might not be independent" with no context. Always tie your answer to the specific scenario in the question — examiners are looking for application, not a recited definition.

Skipping the "check" step in approximations

Even when told to use a Poisson approximation, briefly stating "n is large, p is small" is often worth a method mark on its own — don't skip straight to the calculation.

What examiners reward Clear structure (define the variable and its distribution first), correctly rescaled parameters, careful conversion of inequalities, and — for "explain" or "criticise" questions — answers written in the actual context of the problem rather than generic bookwork.

Also in the full note

Poisson Mean & Variance
Exam Tips & Common Traps

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