IB Maths AA SL Topic 4 — Probability Distributions Paper 1 & 2 ~9 min read

The Binomial Distribution

Flip a coin 20 times — how many heads? Take 50 random people — how many are left-handed? Whenever you’re counting successes in a fixed number of tries with a constant chance, you’ve got a binomial distribution. Learn the 4 conditions, write X ∼ B(n, p), and your GDC does the rest.

📘 What you need to know

Notation: X ∼ B(n, p) means “X is binomially distributed with n trials and probability p of success”.
A binomial model needs 4 conditions: fixed n trials · independent · two outcomes · constant p.
Mean: E(X) = np (in formula booklet)
Variance: Var(X) = np(1 − p) (in formula booklet) · standard deviation = √Var(X)
Use your GDC to calculate any binomial probability — never the formula by hand unless asked.
If p < 0.5 → graph leans right. If p = 0.5 → symmetric. If p > 0.5 → graph leans left.

When is something binomial?

Don’t overthink it. If a problem says “flip a coin n times and count heads” or “survey n people and count how many own a car” — it’s binomial. You’re counting successes across a fixed number of identical, independent attempts.

Formally, X follows a binomial distribution if all 4 conditions hold:

Fixed number of trials

You know n in advance — and it’s a finite, set number.

e.g. flip a coin exactly 20 times ✓
“keep flipping until heads” ✗

Independent trials

Each trial doesn’t affect the others.

e.g. separate coin flips ✓
drawing cards without replacement ✗

Two outcomes only

Each trial is “success” or “failure” — only two results.

e.g. heads/tails, yellow car / not yellow ✓
shoe size, dice number ✗

Constant probability p

p stays the same in every trial.

e.g. fair coin → p = 0.5 each time ✓
swimmer getting tired → p drops ✗

🧠

Memory trick: “BINS”

Binary outcomes (2) · Independent · Number of trials fixed · Same probability. If a scenario fails ANY of the four, it’s not binomial.

The notation: X ∼ B(n, p)

Once you’ve checked the 4 conditions, write the random variable down properly. This is the first thing examiners look for on a binomial question.

X ∼ B(n, p)

Xthe random variable (the count of successes)

nthe number of trials

pthe probability of success on a single trial

So if you flip a fair coin 20 times and let X = number of heads:

X ∼ B(20, 0.5)

Always define your variable in words first. “Let X be the number of heads in 20 flips of a fair coin.” Then write X ∼ B(20, 0.5). That single sentence often earns you 2 marks.

How to set up a binomial model

This is the bread-and-butter exam skill. Here’s the method:

📍

The 4-step setup

1. Identify what counts as a trial. 2. Decide what counts as success. 3. Pull out n (how many trials) and p (chance of success). 4. Define X in words and write X ∼ B(n, p).

Worked-through setup example

Question: “30% of people in a city have blue eyes. A random sample of 30 people is taken. X = number with blue eyes.”

Trial: checking one person’s eye colour.
Success: the person has blue eyes.
n: 30 people · p: 0.3
Write it: X ∼ B(30, 0.3) ✓

🤔 What about “without replacement” — is that still binomial?

Strictly, sampling without replacement breaks the independence rule. But if the population is large and the sample is small relative to it, the change in probability between picks is so tiny that we treat it as binomial. (E.g. 30 people from a whole city — fine. 5 cards from a 52-card deck — NOT fine.)

Spot the binomial (and the fakes)

Examiners love to test whether you can recognise when a binomial doesn’t fit. If a scenario fails any one of the 4 conditions, you can’t use B(n, p).

✅ BINOMIAL

Number of heads in 50 fair coin flips · B(50, 0.5)
Number of 6’s rolled in 30 dice rolls · B(30, 1/6)
Number of yellow cars in a sample of 100 cars (from a big city) · B(100, p)
Number of correctly-guessed multiple choice answers (4 options) on a 25-question test · B(25, 0.25)
Number of people in a sample of 50 who are vaccinated, given 80% vaccinated population · B(50, 0.8)

❌ NOT BINOMIAL

Number of times a coin is flipped until the first head — n is not fixed
Number of caramels eaten when picking 5 from a bag of 6 caramels and 4 marshmallows — not independent
The number a fair dice lands on — more than 2 outcomes
Number of laps under 60s when a swimmer does 50 laps — p drops as they tire
Emails received in an hour — no fixed number of trials

Mean, variance & standard deviation

Two short formulas, both in the formula booklet. Memorise them — they get used in nearly every binomial question.

Mean (expected value)

E(X) = np

✓ in formula booklet

Variance · take √ for standard deviation

Var(X) = np(1 − p)

✓ in formula booklet

Quick example: if X ∼ B(50, 0.08) (50 people, each with 8% chance of being immune to a virus):

E(X) = 50 × 0.08 = 4 people expected to be immune.
Var(X) = 50 × 0.08 × 0.92 = 3.68
SD = √3.68 ≈ 1.92 people

The mean np doesn’t have to be a whole number. E(X) = 4.5 just means on average about 4 or 5 — over many repeats, the long-run average is 4.5.

What does the distribution look like?

You won’t usually have to draw it, but knowing the shape helps you sanity-check answers.

Shape changes with p (when n = 10)

p < 0.5 → most outcomes near 0, long tail to the right.
p = 0.5 → symmetric distribution centred at n/2.
p > 0.5 → most outcomes near n, long tail to the left.

Worked examples

WE 1

Decide if the scenario is binomial

For each of the following, state with reasons whether X follows a binomial distribution.

(a) A fair coin is flipped 30 times. X = number of heads.
(b) A bag has 5 red and 5 blue marbles. 4 are drawn without replacement. X = number of red marbles drawn.
(c) A spinner has sectors red, blue, green. It is spun 20 times. X = number of times it lands on green.

Check the 4 conditions: fixed n, independent, two outcomes, constant p.part (a) — coin flips Fixed n = 30 ✓ Independent ✓ 2 outcomes (H/T) ✓ p = 0.5 constant ✓ YES — X ∼ B(30, 0.5)part (b) — without replacement Drawing without replacement → trials NOT independent. P(red) changes after each draw. NOT binomial — fails independencepart (c) — spinner Trial = “lands on green or not green” → 2 outcomes ✓ Fixed n = 20 ✓ Independent spins ✓ p = 1/3 constant ✓ YES — X ∼ B(20, 1/3) “more than 2 colours” doesn’t kill binomial — just rephrase as success/not-success!

WE 2

Set up a binomial model and find the expected value

It is known that 8% of a large population are immune to a particular virus. Mark takes a random sample of 50 people from this population.

(a) State the distribution that Mark uses for the number of immune people.
(b) State two assumptions Mark must make.
(c) Calculate the expected number of immune people in the sample.

Define X, identify n and p, then apply E(X) = np.part (a) — define the distribution Trial: check if a person is immune. Success: person is immune. n = 50, p = 0.08. Let X = number immune. X ∼ B(50, 0.08)part (b) — assumptions 1. Each person has the SAME 8% chance of being immune (constant p). 2. The sample is random → people are independent (one person’s immunity doesn’t affect another’s). e.g. if all 50 came from one family, independence would fail.part (c) — expected value Apply E(X) = np: E(X) = 50 × 0.08 = 4 Expected ≈ 4 people immune always state assumptions in WORDS — examiners want the reasoning, not just the formula!

WE 3

Find mean, variance, standard deviation

The random variable Y ∼ B(80, 0.25). Find:

(a) E(Y) (b) Var(Y) (c) the standard deviation of Y, to 3 s.f.

Use formulas E(Y) = np and Var(Y) = np(1−p) directly.part (a) — mean E(Y) = 80 × 0.25 = 20 E(Y) = 20part (b) — variance Var(Y) = 80 × 0.25 × (1 − 0.25) = 80 × 0.25 × 0.75 = 15 Var(Y) = 15part (c) — sd SD = √15 = 3.872983… SD ≈ 3.87 (3 s.f.) SD is just √variance — never forget to take the root!

WE 4

Use E(X) to find n

The random variable X ∼ B(n, 0.4) and E(X) = 18. Find n and the variance of X.

Use E(X) = np and rearrange to solve for n. Set up equation: np = 18 → n × 0.4 = 18 n = 18 ÷ 0.4 = 45 n = 45 Now find variance: Var(X) = np(1−p) = 45 × 0.4 × 0.6 = 10.8 Var(X) = 10.8 when given E(X), just rearrange np = E(X) — easy mark!

WE 5

Use mean and variance to find n and p

The random variable X ∼ B(n, p) has mean 12 and variance 4.8. Find n and p.

Two equations, two unknowns. Divide them to get p. Equation 1: np = 12 Equation 2: np(1 − p) = 4.8 Divide eq 2 by eq 1: np(1 − p)np = 4.812 1 − p = 0.4 → p = 0.6 Sub back into np = 12: n × 0.6 = 12 → n = 20 n = 20, p = 0.6 dividing variance by mean kills n and gives you (1 − p) directly — slick trick!

💡 Top tips

Always check the 4 conditions (BINS) before declaring a variable binomial. State them clearly in part (a) of any modelling question.
Define X in words first, then write X ∼ B(n, p). It earns method marks even if the rest goes wrong.
Use your GDC for actual probabilities — don’t compute ⁿC_x by hand unless asked to “show working”.
Memorise: E(X) = np and Var(X) = np(1 − p). Both are in the formula booklet but you won’t have time to flip pages.
Standard deviation = √variance. Easy mark missed when students forget the square root.
“Without replacement” with a small population → not binomial. With a large population → fine to use binomial.
If given E(X) and Var(X), divide them: ^Var⁄_Mean = 1 − p. Quick way to find p.
Sanity check the mean — for X ∼ B(50, 0.08), E(X) = 4 should “feel right” given 8% of 50.

⚠ Common mistakes

Calling something binomial when it isn’t. “Until first success” or “without replacement (small pop)” both fail — re-check BINS every time.
Forgetting to define X. Don’t just write X ∼ B(50, 0.08) — say what X represents in words first.
Mixing up n and p. Always: B(n, p) — number of trials FIRST, probability second.
Using p as a percentage. 8% means p = 0.08, not p = 8. Convert before substituting.
Forgetting (1 − p) in variance. Var = np(1 − p), NOT np.
Reporting variance as standard deviation. SD requires a square root at the end.
Saying E(X) must be a whole number. It can be 4.5, 7.3, etc. — it’s a long-run average, not an actual count.
Using “more than 2 outcomes” as automatic disqualification. If you can rephrase the trial as success/not-success (e.g. yellow vs not-yellow), it’s still binomial.

Now you can recognise a binomial scenario, write X ∼ B(n, p), and find E(X) and Var(X) in seconds. The next note shows you how to calculate actual binomial probabilities like P(X = 5), P(X ≤ 10), and P(8 < X < 15) — all using your GDC.

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