Binomial and Poisson distributions

IB Mathematics: Analysis and Approaches HL· Topic 4: Statistics and probability, 4.6 Discrete distributions· 30 min read

1. Binomial Distribution★★☆☆☆⏱ 15 min

✓ Calculator OK

📘 Definition

Binomial Distribution

$X \sim B (n, p)$

A discrete probability distribution for the number of successes in $n$ independent Bernoulli trials, where each trial has two outcomes and constant success probability $p$ .

Example:

Number of correct multiple choice answers when guessing 10 questions.

Four conditions must be satisfied for a distribution to be binomial: (1) fixed number of trials, (2) two possible outcomes per trial, (3) trials are independent, (4) constant probability of success. The key properties are:

Probability mass function: $P (X = k) = (k n) p^{k} (1 - p)^{n - k}$
Expected value: $E (X) = n p$
Variance: $Var (X) = n p (1 - p)$

📐 Worked Example

12% of students in a school are left-handed. 7 students are selected at random. Find the probability that exactly 2 are left-handed, and the expected number of left-handed students.

1
Check conditions: fixed $n = 7$ trials, two outcomes (left/right-handed), independent selection, constant $p = 0.12$ . So $X \sim B (7, 0.12)$
2
Use the binomial PMF for $k = 2$ :
3
$P(X=2) = \binom{7}{2} (0.12)^2 (0.88)^5$
4
Calculate: $(2 7) = 21$ , so $21 \times 0.0144 \times 0.4182 \approx 0.160
5
Calculate expected value:
6
$E(X) = np = 7 \times 0.12 = 0.84$

Exam tip:

Always verify the four binomial conditions first to earn full method marks.

2. Poisson Distribution★★★☆☆⏱ 15 min

✓ Calculator OK

📘 Definition

Poisson Distribution

$X \sim Po (λ)$

A discrete probability distribution for the number of events occurring in a fixed interval of time or space, when events occur at a constant average rate and independently of each other.

Example:

Number of cars passing a checkpoint in 1 hour.

A unique property of the Poisson distribution is that its mean equals its variance. The key properties are:

Probability mass function: $P (X = k) = \frac{e ^{- λ} λ ^{k}}{k !}$
Expected value: $E (X) = λ$
Variance: $Var (X) = λ$

📐 Worked Example

A website receives an average of 4 hits per 10 minutes. Find the probability of exactly 6 hits in 10 minutes, and the probability of fewer than 2 hits.

1
Confirm conditions: constant average rate, independent events, so $X \sim Po (4)$ with $λ = 4$
2
Calculate $P (X = 6)$ :
3
$P(X=6) = \frac{e^{-4} 4^6}{6!} = \frac{4096 e^{-4}}{720} \approx 0.104$
4
Calculate $P (X < 2) = P (X = 0) + P (X = 1)$ :
5
$P(X < 2) = \frac{e^{-4} 4^0}{0!} + \frac{e^{-4} 4^1}{1!} = e^{-4}(1 + 4) = 5e^{-4} \approx 0.092$

3. Poisson Approximation to Binomial★★★☆☆⏱ 20 min

✓ Calculator OK

When the binomial distribution has a large number of trials $n$ and small success probability $p$ , it can be approximated by a Poisson distribution with parameter $λ = n p$ . The IB rule of thumb for this approximation to be valid is $n > 10$ and $p < 0.1$ , or $n > 50$ and $n p < 10$ .

📐 Worked Example

0.5% of products made by a factory are defective. 200 products are selected at random. Use a Poisson approximation to find the probability of exactly 2 defective products.

1
Confirm conditions: $n = 200$ large, $p = 0.005$ small. Calculate $λ = n p = 200 \times 0.005 = 1$
2
Approximate $X \sim B (200, 0.005)$ with $X \approx Po (1)$
3
Calculate $P (X = 2)$ using Poisson PMF:
4
$P(X=2) = \frac{e^{-1} 1^2}{2!} = \frac{e^{-1}}{2} \approx 0.184$

✓ Quick check

When can you use Poisson approximation to a binomial distribution $X \sim B (100, 0.02)$ ?
- Never
- Yes, because n is large and p is small
- Only if n < 50
- Only if p > 0.1
Reveal answer
1 —
Correct: $n = 100$ is large, $p = 0.02$ is small, so Poisson approximation is appropriate here, with $λ = 2$ .

4. Common Pitfalls

Wrong move:

Assuming any count data automatically follows a Poisson distribution.

Why:

Poisson requires events to occur independently at a constant rate. Clustered or dependent events violate this assumption.

Correct move:

Always check the distribution conditions before starting calculations to earn method marks even if your final answer is incorrect.

Wrong move:

Forgetting to scale $λ$ when changing the interval size for a Poisson distribution.

Why:

$λ$ is the mean rate for the given interval, so it scales proportionally with interval size.

Correct move:

Multiply $λ$ by the scaling factor for the new interval, e.g., 2 per 10 minutes = 6 per 30 minutes.

Wrong move:

Using $n p$ for binomial variance, instead of $n p (1 - p)$ .

Why:

Variance for binomial depends on both n and p, and is always less than the mean when $p > 0$ .

Correct move:

Remember: for binomial, variance = mean $\times (1 - p)$ , so $Var (X) = n p (1 - p)$ .

Wrong move:

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

Why:

The approximation only works when p is small. For large p, you can model failures instead, or use a normal approximation.

Correct move:

Only use Poisson approximation for binomial when $p < 0.1$ and n is large, per IB requirements.

5. Quick Reference Cheatsheet

Distribution	Notation	Mean	Variance	Use Case
Binomial	$X \sim B (n, p)$	$n p$	$n p (1 - p)$	successes in n fixed trials
Poisson	$X \sim Po (λ)$	$λ$	$λ$	events in fixed interval
Poisson approx to Binomial	$X \approx Po (n p)$	$n p$	$n p$	n large, $p < 0.1$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · P1
Binomial probability calculation
2024 · P2
Poisson approximation question
2023 · P1
Compare binomial and Poisson

Going deeper

syllabusIB AA HL Syllabus Topic 4

What's Next

Binomial and Poisson distributions are core discrete probability models that you will use frequently in both Paper 1 and Paper 2 IB AA HL exams, often combined with conditional probability or hypothesis testing for population parameters. Mastering the conditions for selecting the correct distribution is a key skill that will carry over to continuous distributions, which are the next major topic in the statistics unit. You will now build on this foundation to study the most common continuous distribution, the normal distribution, which is also used to approximate binomial distributions for large n.

1. Binomial Distribution★★☆☆☆⏱ 15 min

Binomial Distribution

2. Poisson Distribution★★★☆☆⏱ 15 min

Poisson Distribution

3. Poisson Approximation to Binomial★★★☆☆⏱ 20 min

4. Common Pitfalls

5. Quick Reference Cheatsheet

successes in n fixed trials

events in fixed interval

When this came up on past exams

Going deeper

What's Next