# The Binomial and Poisson Distributions (Edexcel IAL S2)

> Edexcel International A-Level Mathematics · IAL S2 2018
> Source: https://www.owlsprep.com/study/edexcel-ial-math-s2-the-binomial-and-poisson-distributions/

This guide covers all Edexcel IAL S2 content for binomial and Poisson distributions, including model appropriateness checks, cumulative probability calculations, Poisson additive properties, and Poisson approximation to the binomial, aligned with the 2018 specification.

**Prerequisites:** [Edexcel IAL S1 Discrete Probability Distributions](https://www.owlsprep.com/study/edexcel-ial-math-s1-discrete-probability-distributions/); [Basic cumulative probability calculations](https://www.owlsprep.com/study/edexcel-ial-math-s1-cumulative-probability/)

## Learning objectives

- Identify when binomial or Poisson distribution is appropriate for real-world scenarios
- Calculate cumulative probabilities for both distributions via direct calculation or statistical tables
- Apply the additive property of the Poisson distribution for scaling time/space intervals
- Use the given mean and variance formulae for binomial and Poisson distributions
- Apply Poisson as an approximation to the binomial distribution when conditions are met

## Binomial & Poisson Distribution Model Appropriateness

**Binomial Distribution** — 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.

*Notation:* $X \sim B(n,p)$

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

**Poisson Distribution** — 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.

*Notation:* $X \sim Po(\lambda)$

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

> **info**
>
> When commenting on model appropriateness in exams, you must explicitly reference which conditions are met or violated for the scenario given to earn full marks.

**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. 2. Check constant success probability: If absence reasons are independent (e.g. individual illness), p=0.03 is constant: condition is partially met.
3. 3. Check independent trials: If absences are due to contagious illness, trials are not independent: this is a key potential limitation.
4. 4. Final conclusion: The model is appropriate if absences are independent, and inappropriate if there is a widespread illness causing multiple related absences.

*Calculator:* allowed

## Cumulative Probability Calculations

Cumulative probabilities for $B(n,p)$ and $Po(\lambda)$ 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.

> **tip**
>
> If you need $P(X > x)$, use the complement rule: $P(X > x) = 1 - P(X \leq x)$. For $P(X \geq x)$, use $1 - P(X \leq x-1)$.

**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: $P(X=k) = \binom{10}{k} 0.2^k 0.8^{10-k}$
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. 2. Sum the terms: $0.3020 + 0.2013 + 0.0881 = 0.5914$
6. 3. 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.

*Calculator:* allowed

## Distribution Properties & Poisson Additive Rule

Mean and variance for both distributions are given in the formula booklet (no derivation required):
- For $X \sim B(n,p)$: $E(X) = np$, $Var(X) = np(1-p)$
- For $Y \sim Po(\lambda)$: $E(Y) = Var(Y) = \lambda$

**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 $Po(k\lambda)$. 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, $X \sim Po(6)$
2. 2. We need $P(X < 2) = P(X \leq 1)$
3. 3. Use cumulative Poisson tables for $\lambda=6$: $P(X \leq 1) = 0.0174$

*Calculator:* allowed

## Poisson Approximation to the Binomial Distribution

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 $\lambda = np$.

> **warning**
>
> Edexcel examiners expect you to explicitly state the conditions for Poisson approximation to the binomial before using it in a solution to earn full method marks.

**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. 2. Check conditions for Poisson approximation: $n=100 \geq 50$, $p=0.02 \leq 0.1$: conditions are satisfied
3. 3. Calculate $\lambda = np = 100 \times 0.02 = 2$, so the approximate distribution is $X \approx Po(2)$
4. 4. We need $P(X < 3) = P(X \leq 2)$
5. 5. Use cumulative Poisson tables for $\lambda=2$: $P(X \leq 2) = 0.6767$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to state conditions for model appropriateness or Poisson approximation
  - Why it fails: Examiners allocate explicit marks for justifying choices, so you lose method marks even if calculations are correct
  - Correct: 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:** Using the wrong cumulative probability complement, e.g. calculating $P(X \geq 3)$ as $1 - P(X \leq 3)$
  - Why it fails: You exclude X=3 incorrectly, leading to wrong final answer
  - Correct: 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:** Applying the Poisson additive property to non-independent intervals, or scaling incorrectly
  - Why it fails: The additive property only holds when intervals are independent and the event rate is constant across intervals
  - Correct: Only scale λ if intervals are identical, independent, and the average rate does not change between intervals
- **Wrong:** Confusing variance formulae for binomial and Poisson distributions
  - Why it fails: Mistakes in variance calculation lead to incorrect parameter values for approximations or follow-up questions
  - Correct: Remember: Binomial variance = $np(1-p)$, Poisson variance = $\lambda =$ mean; cross-check with the formula booklet if unsure
- **Wrong:** Using Poisson approximation for binomial when p is large (e.g. p=0.8)
  - Why it fails: Poisson approximation is only valid for small p; large p cases use normal approximation (separate S2 topic)
  - Correct: Only use Poisson approximation when $p \leq 0.1$ and $n \geq 50$ per Edexcel guidance

## Cheatsheet

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

## 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.

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