The Binomial and Poisson Distributions (Edexcel IAL S2)

Q: When can I use Poisson to approximate a binomial distribution?

Use Poisson approximation for if n is large () and p is small (), so that is moderate (typically , per Edexcel guidance).

Q: Do I need to derive the mean and variance for binomial or Poisson?

No, derivations are not required for Edexcel IAL S2. You can directly use the formulae given in the official formula booklet: , for ; for .

Edexcel International A-Level Mathematics· S2 §1.1 to §1.3· 25 min read

1. Binomial & Poisson Distribution Model Appropriateness★★☆☆☆⏱ 5 min

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📘 Definition

Binomial Distribution

$X \sim B (n, p)$

Discrete distribution modeling the number of successes in n independent, identical Bernoulli trials, each with constant success probability p. Conditions: Fixed n, independent trials, 2 outcomes (success/failure), constant p.

Example:

Number of defective items in a batch of 20, where 5% of items are defective.

📘 Definition

Poisson Distribution

$X \sim P o (λ)$

Discrete distribution modeling the number of events occurring in a fixed interval of time/space, where events occur independently at a constant average rate. Conditions: Events occur independently, at constant average rate, no simultaneous events.

Example:

Number of customers arriving at a café per hour, with average 8 arrivals per hour.

📐 Worked Example

A teacher models the number of students absent from a class of 30 on a random school day as a binomial distribution, with probability of absence 0.03 per student. Comment on the appropriateness of this model.

1
1. Test binomial conditions against the scenario: Fixed n=30 trials (students), 2 outcomes (absent/present): both conditions are met.
2
1. Check constant success probability: If absence reasons are independent (e.g. individual illness), p=0.03 is constant: condition is partially met.
3
1. Check independent trials: If absences are due to contagious illness, trials are not independent: this is a key potential limitation.
4
1. Final conclusion: The model is appropriate if absences are independent, and inappropriate if there is a widespread illness causing multiple related absences.

2. Cumulative Probability Calculations★★★☆☆⏱ 7 min

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Cumulative probabilities for $B (n, p)$ and $P o (λ)$ can be calculated directly using the probability mass function (pmf) from the formula booklet, or looked up in the provided cumulative statistical tables to save time in exams. Tables give $P (X \leq x)$ for specified parameter values.

📐 Worked Example

Given $X \sim B (10, 0.2)$ , calculate $P (2 \leq X \leq 4)$ first via direct calculation, then verify using cumulative tables.

1
1. Direct calculation requires summing the pmf for x=2, 3, 4: $P (X = k) = (k 10) 0. 2^{k} 0. 8^{10 - k}$
2
$P(X=2) = \binom{10}{2} 0.2^2 0.8^8 = 45 \times 0.04 \times 0.16777216 = 0.3020$
3
$P(X=3) = \binom{10}{3} 0.2^3 0.8^7 = 120 \times 0.008 \times 0.2097152 = 0.2013$
4
$P(X=4) = \binom{10}{4} 0.2^4 0.8^6 = 210 \times 0.0016 \times 0.262144 = 0.0881$
5
1. Sum the terms: $0.3020 + 0.2013 + 0.0881 = 0.5914$
6
1. Verify with tables: $P (X \leq 4) = 0.9672$ , $P (X \leq 1) = 0.3758$ , so $0.9672 - 0.3758 = 0.5914$ , matching the direct calculation.

3. Distribution Properties & Poisson Additive Rule★★★☆☆⏱ 6 min

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Mean and variance for both distributions are given in the formula booklet (no derivation required):

For $X \sim B (n, p)$ : $E (X) = n p$ , $V a r (X) = n p (1 - p)$
For $Y \sim P o (λ)$ : $E (Y) = V a r (Y) = λ$

📘 Definition

Poisson Additive Property

If events occur at a constant average rate of λ per unit interval, then the number of events in k identical independent intervals follows $P o (k λ)$ . This only applies when intervals are independent and the rate is constant.

📐 Worked Example

A bakery receives an average of 3 customer complaints per week. Assuming complaints follow a Poisson distribution, find the probability that the bakery receives fewer than 2 complaints in a 2-week period.

1
1. Apply the additive property: Rate per 2 weeks = $2 \times 3 = 6$ , so let $X$ = number of complaints in 2 weeks, $X \sim P o (6)$
2
1. We need $P (X < 2) = P (X \leq 1)$
3
1. Use cumulative Poisson tables for $λ = 6$ : $P (X \leq 1) = 0.0174$

4. Poisson Approximation to the Binomial Distribution★★★★☆⏱ 7 min

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For binomial distributions where n is large ( $n \geq 50$ ) and p is small ( $p \leq 0.1$ ), the Poisson distribution provides a simpler approximation. The Poisson parameter λ is set equal to the mean of the binomial distribution, so $λ = n p$ .

📐 Worked Example

A factory produces components, 2% of which are defective. A random sample of 100 components is selected. Use a Poisson approximation to find the probability that fewer than 3 components are defective.

1
1. Define the original binomial variable: Let $X$ = number of defective components, $X \sim B (100, 0.02)$
2
1. Check conditions for Poisson approximation: $n = 100 \geq 50$ , $p = 0.02 \leq 0.1$ : conditions are satisfied
3
1. Calculate $λ = n p = 100 \times 0.02 = 2$ , so the approximate distribution is $X \approx P o (2)$
4
1. We need $P (X < 3) = P (X \leq 2)$
5
1. Use cumulative Poisson tables for $λ = 2$ : $P (X \leq 2) = 0.6767$

5. Common Pitfalls

Wrong move:

Forgetting to state conditions for model appropriateness or Poisson approximation

Why:

Examiners allocate explicit marks for justifying choices, so you lose method marks even if calculations are correct

Correct move:

Always reference the 4 binomial conditions or 3 Poisson conditions when commenting on appropriateness, and state $n \geq 50$ , $p \leq 0.1$ when using Poisson approximation to binomial

Wrong move:

Using the wrong cumulative probability complement, e.g. calculating $P (X \geq 3)$ as $1 - P (X \leq 3)$

Why:

You exclude X=3 incorrectly, leading to wrong final answer

Correct move:

Use $P (X \geq k) = 1 - P (X \leq k - 1)$ , and $P (X > k) = 1 - P (X \leq k)$ for all discrete distributions

Wrong move:

Applying the Poisson additive property to non-independent intervals, or scaling incorrectly

Why:

The additive property only holds when intervals are independent and the event rate is constant across intervals

Correct move:

Only scale λ if intervals are identical, independent, and the average rate does not change between intervals

Wrong move:

Confusing variance formulae for binomial and Poisson distributions

Why:

Mistakes in variance calculation lead to incorrect parameter values for approximations or follow-up questions

Correct move:

Remember: Binomial variance = $n p (1 - p)$ , Poisson variance = $λ =$ mean; cross-check with the formula booklet if unsure

Wrong move:

Using Poisson approximation for binomial when p is large (e.g. p=0.8)

Why:

Poisson approximation is only valid for small p; large p cases use normal approximation (separate S2 topic)

Correct move:

Only use Poisson approximation when $p \leq 0.1$ and $n \geq 50$ per Edexcel guidance

6. Quick Reference Cheatsheet

Concept	Notation / Formula	Key Conditions / Notes
Binomial Distribution	$X \sim B (n, p)$ ; $E (X) = n p$ , $V a r (X) = n p (1 - p)$	Fixed n, independent trials, 2 outcomes, constant p; use tables for cumulative probs
Poisson Distribution	$X \sim P o (λ)$ ; $E (X) = V a r (X) = λ$	Events independent, constant rate, no simultaneous events; use tables for cumulative probs
Poisson Additive Property	Rate per k intervals = $k λ$ , $X \sim P o (k λ)$	Only for independent, identical intervals with constant rate
Poisson Approx to Binomial	$λ = n p$ , $X \approx P o (n p)$	Valid if $n \geq 50$ , $p \leq 0.1$ ; simplifies large n, small p binomial calculations

7. Frequently Asked

When can I use Poisson to approximate a binomial distribution?

Use Poisson approximation for $B (n, p)$ if n is large ( $n \geq 50$ ) and p is small ( $p \leq 0.1$ ), so that $n p$ is moderate (typically $n p < 10$ , per Edexcel guidance).

Do I need to derive the mean and variance for binomial or Poisson?

No, derivations are not required for Edexcel IAL S2. You can directly use the formulae given in the official formula booklet: $E [X] = n p$ , $V a r (X) = n p (1 - p)$ for $B (n, p)$ ; $E [Y] = V a r (Y) = λ$ for $P o (λ)$ .

Going deeper

formula_bookletEdexcel IAL Maths Formulae & Statistical TablesProvided in all S2 exams

What's Next

Now that you have mastered binomial and Poisson distributions for Edexcel IAL S2, you are ready to move to more advanced statistics topics that build on these core probability models. The next key topic in S2 is continuous probability distributions, including the normal distribution, which you will use for approximations and hypothesis testing later in the unit. You will also apply these discrete distribution models to hypothesis testing for binomial and Poisson parameters in S2 Section 4, where you will learn to test claims about population proportions and event rates. Ensure you practice past paper questions focused on this topic to familiarize yourself with Edexcel's exam phrasing and mark scheme expectations, especially for questions asking you to comment on model appropriateness, which are common high-mark questions. Don't forget to use the provided formula booklet and statistical tables during practice to simulate exam conditions.