IB Maths AI HLBinomial DistributionPaper 1 & 2~7 min read
The Binomial Distribution
The binomial distribution counts the number of successes in a fixed number of independent yes/no trials. If a situation passes four conditions — fixed trials, independence, two outcomes, constant probability — you can model it with X ∼ B(n, p) and pull its mean and variance straight from two booklet formulae. The exam skill is half spotting when the model applies, and half setting it up cleanly.
📘 What you need to know
Notation: X ∼ B(n, p) — n = number of trials, p = probability of success. Failure probability is 1 − p (often q).
Four conditions: fixed number of trials, independent trials, exactly two outcomes, constant probability of success.
“Success” is just a label — it doesn’t have to be a “good” outcome.
Probability formula: P(X = r) = nCrpr(1 − p)n−r — but use your GDC in practice.
Mean: E(X) = np (in the booklet).
Variance: Var(X) = np(1 − p) (in the booklet); square-root it for the standard deviation.
Shape: p < 0.5 tails right, p > 0.5 tails left, p = 0.5 symmetric.
The four conditions
A discrete random variable X follows a binomial distribution if it counts successes in an experiment meeting all four conditions. A handy mnemonic is “BINS”.
🧠 Memory aid — “BINS”
Binary (exactly two outcomes: success / failure) · Independent trials · N fixed number of trials · Same probability p every trial. Miss any one and it’s not binomial.
Binomial ✓
fits BINS
Number of tails in 20 fair coin flips: B(20, 0.5). Fixed trials, independent, two outcomes, constant p.
Not binomial ✗
breaks a rule
Caramels eaten from a bag (no replacement → not independent, p changes). Or rolls of a die (more than two outcomes).
🤔 “More than two colours” — can it still be binomial?
Yes, if you frame the trial as a yes/no. Counting yellow cars in a car park of 100 has many possible colours, but the trial is “yellow or not yellow” — two outcomes. As long as the other three conditions hold, it’s binomial. Sampling without replacement from a large population also counts: each pick is approximately independent with constant p.
Scenario
Why it fails
Emails received in an hour
Number of trials not fixed / infinite
Flips until first heads
Number of trials not fixed
Caramels eaten (no replacement)
Trials not independent; p changes
Shoe size; die roll
More than two outcomes
Pool lengths swum under a minute
p drops as the swimmer tires
Setting up a model
Once you’ve checked the conditions, setting up the model is a short, repeatable routine. Always state your random variable in words.
🧭 Recipe — setting up a binomial model
Identify a trial — e.g. checking whether a person is immune.
Define “success” — e.g. the person is immune (just a label).
Find the parameters — n = number of trials, p = probability of success.
State the variable — “Let X be the number of … ” then write X ∼ B(n, p).
Exam habit: when a question mixes several distributions, make it clear which one each variable follows. Naming the distribution explicitly (e.g. X ∼ B(50, 0.08)) is often worth a mark on its own.
Mean, variance & shape
Two booklet formulae give the centre and spread of a binomial distribution directly from n and p — no table needed.
Mean and variance of B(n, p)
E(X) = np Var(X) = np(1 − p)
both in the formula booklet ✓
How p controls the shape
Small p bunches the bars on the left (tail right); large p bunches them right (tail left).
Worked examples
WE 1
State the distribution
It is known that 8% of a large population are immune to a virus. Mark samples 50 people and models the number immune with a binomial distribution. State the distribution.
trial: check if a person is immunesuccess = the person is immune.parameters: n = 50, p = 0.08Let X = number of immune people in the sample.X ∼ B(50, 0.08)
WE 2
State the assumptions
State two assumptions Mark must make to use a binomial model.
1. constant probabilityeach person has an 8% chance of being immune.2. independencethe sample is random and people are independent — one person being immune doesn’t affect others (e.g. not all 50 from one family).constant p AND independent trials
WE 3
Expected number (mean)
For Mark’s model X ∼ B(50, 0.08), calculate the expected number of immune people.
E(X) = np= 50 × 0.08 = 4E(X) = 4 peoplethe mean comes straight from the booklet formula.
WE 4
Variance and standard deviation
For the same model X ∼ B(50, 0.08), find the variance and standard deviation.
Var(X) = np(1 − p)= 50 × 0.08 × 0.92 = 3.68sd = √Var(X)= √3.68 = 1.918…Var = 3.68, sd ≈ 1.92remember 1 − p = 0.92; the sd is just the square root.
WE 5
Is it binomial?
A person eats 5 sweets from a bag of 6 caramels and 4 marshmallows. Let C = number of caramels eaten. Can C be modelled binomially? Explain.
check the conditionsfixed trials (5) ✓, two outcomes (caramel / not) ✓but: no replacementeating one caramel changes what’s left, so p is NOT constant and trials are NOT independent.No — not binomialsampling without replacement from a small bag breaks two conditions.
💡 Top tips
Check all four conditions (BINS) before declaring a model binomial.
Define the variable in words, then write X ∼ B(n, p).
E(X) = np and Var(X) = np(1 − p) — both in the booklet.
“Success” is a label — it can be a “bad” outcome like being infected.
Large population sampling can be treated as binomial even without replacement.
Round the mean sensibly if a question asks for a number of people (but keep exact values for further working).
⚠ Common mistakes
Calling it binomial when trials aren’t independent or p changes (no-replacement, tiring swimmer).
Forgetting the 1 − p in the variance — it’s np(1 − p), not np.
Using variance instead of sd (or vice versa) — square-root for the standard deviation.
Not stating the random variable, losing easy “set-up” marks.
Mixing up n and p in the notation B(n, p).
Assuming “more than two colours” rules out binomial — reframe as the yes/no trial.
Next up — Calculating Binomial Probabilities. You’ll use your GDC’s binomial PD and CD functions to find P(X = x) and cumulative probabilities like P(a ≤ X ≤ b), plus the integer tricks for turning strict inequalities into ones your calculator can handle.
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