Binomial, Poisson and normal distributions

IB Mathematics: Applications and Interpretation HL· 5.6 Common probability distributions· 25 min read

1. Binomial Distribution★★☆☆☆⏱ 7 min

📘 Definition

Binomial Distribution

$X \sim B (n, p)$

A discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with constant probability of success.

Example:

Number of heads in 10 independent coin flips

The probability of getting exactly $k$ successes is given by the probability mass function:

P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

The expected value (mean) and variance of a binomial distribution are:

E(X) = np, \quad Var(X) = np(1-p)

📐 Worked Example

5% of light bulbs produced at a factory are defective. A random sample of 10 bulbs is selected. What is the probability that exactly 2 bulbs are defective?

1
Check binomial conditions: fixed $n = 10$ trials, two outcomes (defective/not defective), constant $p = 0.05$ , independent trials (population large), so binomial is appropriate.
2
Substitute $k = 2$ into the PMF:
3
$P(X=2) = \binom{10}{2} (0.05)^2 (0.95)^8$
4
Calculate $(2 10) = 45$ , then simplify to get:
5
$P(X=2) \approx 0.0746$

Exam tip:

Marks are often awarded for justifying why you chose a binomial distribution, always confirm the four conditions in your working.

2. Poisson Distribution★★★☆☆⏱ 6 min

📘 Definition

Poisson Distribution

$X \sim P o (λ)$

A discrete probability distribution that models the number of events occurring in a fixed interval of time/space, when events occur at a constant known mean rate and independently of each other.

Example:

Number of customers arriving at a café in 1 hour

The probability of exactly $k$ events occurring is:

P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}

A key property of the Poisson distribution is that its mean and variance are equal to $λ$ :

E(X) = \lambda, \quad Var(X) = \lambda

📐 Worked Example

On average, 3 cars pass through a crosswalk every 10 minutes. What is the probability that exactly 5 cars pass through in a 10-minute interval?

1
Check Poisson conditions: fixed interval, constant mean rate, independent events. So $X \sim P o (λ = 3)$ .
2
Substitute $k = 5$ into the PMF:
3
$P(X=5) = \frac{e^{-3} 3^5}{5!}$
4
Calculate $3^{5} = 243$ , $5! = 120$ , so simplify:
5
$P(X=5) \approx 0.1008$

3. Normal Distribution★★★☆☆⏱ 7 min

📘 Definition

Normal Distribution

$X \sim N (μ, σ^{2})$

A continuous symmetric bell-shaped probability distribution, widely used to model naturally occurring continuous variables.

Example:

Heights of adult humans

The probability density function is:

f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

In IB AI HL, you will use your GDC to calculate cumulative probabilities and quantiles for normal distributions, rather than using the PDF directly. The empirical rule states ~68% of data lies within 1 standard deviation of the mean, ~95% within 2, ~99.7% within 3.

📐 Worked Example

Given $X \sim N (20, 5^{2})$ , find $P (X < 24)$ and the value of $k$ such that $P (X > k) = 0.1$ .

1
First, identify parameters: $μ = 20$ , $σ = 5$ (note that the second parameter is variance, so $σ^{2} = 25$ , $σ = 5$ ).
2
For $P (X < 24)$ , use the cumulative normal function on your GDC:
3
Result: $P (X < 24) \approx 0.7881$
4
For $P (X > k) = 0.1$ , we have $P (X < k) = 0.9$ , use inverse normal:
5
Result: $k \approx 26.41$

Exam tip:

Always check if the question gives variance or standard deviation; mixing these up when entering values into your GDC is a very common mistake.

4. Approximations Between Distributions★★★★☆⏱ 5 min

When calculations for one distribution become tedious (usually for large $n$ ), we can approximate it with another distribution if conditions are met. Two key approximations are taught in IB AI HL:

Poisson approximation to binomial: when $n$ is large ( $n > 10$ ) and $p$ is small ( $p < 0.1$ ), approximate $B (n, p)$ with $P o (λ = n p)$
Normal approximation: for $B (n, p)$ , approximate with $N (n p, n p (1 - p))$ when $n p > 5$ and $n (1 - p) > 5$ ; for $P o (λ)$ , approximate with $N (λ, λ)$ when $λ > 10$

📐 Worked Example

1000 seeds are planted, each germinates with probability 0.002. Use Poisson approximation to find the probability that fewer than 3 seeds germinate.

1
Check conditions: $n = 1000$ large, $p = 0.002$ small, so Poisson approximation is appropriate. $λ = n p = 1000 \times 0.002 = 2$ .
2
We want $P (X < 3) = P (X \leq 2) = P (X = 0) + P (X = 1) + P (X = 2)$
3
$P(X \leq 2) = \frac{e^{-2} 2^0}{0!} + \frac{e^{-2} 2^1}{1!} + \frac{e^{-2} 2^2}{2!} = 5e^{-2}$
4
Simplify to get:
5
$P(X < 3) \approx 0.677$

5. Common Pitfalls

Wrong move:

Forgetting continuity correction when approximating a discrete distribution with normal

Why:

Discrete distributions only take integer values, so omitting the correction gives inaccurate probability

Correct move:

Always adjust the bound by ±0.5 when doing a normal approximation to a discrete distribution

Wrong move:

Mixing up variance and standard deviation when entering normal distribution parameters into GDC

Why:

Notation $X \sim N (μ, σ^{2})$ uses variance as the second parameter, and students often enter the wrong value

Correct move:

Always confirm if the question gives variance or standard deviation, and double-check your input before calculating

Wrong move:

Using binomial distribution when sampling without replacement from a small population

Why:

Trials are not independent, so the probability of success changes between draws

Correct move:

Only use binomial if the population is at least 10 times larger than the sample size

Wrong move:

Using Poisson approximation to binomial when p is large

Why:

Poisson approximation is only accurate when p is very small, it will give large errors for large p

Correct move:

Use normal approximation to binomial when $n p > 5$ and $n (1 - p) > 5$ , regardless of p size

6. Quick Reference Cheatsheet

Distribution	Type	Notation	Mean	Variance	PMF/PDF
Binomial	Discrete	$B (n, p)$	$n p$	$n p (1 - p)$	$(k n) p^{k} (1 - p)^{n - k}$
Poisson	Discrete	$P o (λ)$	$λ$	$λ$	$\frac{e ^{- λ} λ ^{k}}{k !}$
Normal	Continuous	$N (μ, σ^{2})$	$μ$	$σ^{2}$	$\frac{1}{σ 2 π} e^{- \frac{( x - μ ) ^{2}}{2 σ ^{2}}}$

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 · 1
Binomial probability calculation
2024 · 2
Normal approximation question
2023 · 1
Distribution identification

What's Next

Understanding these three core distributions is foundational for almost all further statistical inference topics you will encounter in IB AI HL. These distributions are used to model a huge range of real-world processes, from quality control in manufacturing to predicting customer foot traffic, and form the backbone of hypothesis testing and confidence interval calculation. Once you are comfortable identifying when to use each distribution, justifying your choice and calculating probabilities, you are ready to move on to more advanced statistical topics that build directly on this knowledge.