IB Maths: The Binomial Distribution

You need to spot when a situation fits the binomial model — fixed trials, independence, two outcomes, constant probability — then apply P(X = x), the mean np, the variance np(1-p), and read probabilities off your GDC. That's genuinely the whole point of this topic for exam purposes.

A binomial distribution needs four things: a fixed number of trials (n), independence between trials, exactly two outcomes per trial, and a constant probability of success (p) throughout. Miss any one — say you're sampling without replacement from a small population — and you can't model it as B(n, p).

The formula is P(X = x) = (n choose x) · p^x · (1-p)^(n-x), giving the probability of exactly x successes in n independent trials. On your GDC that's binompdf(n, p, x) for an exact value, or binomcdf(n, p, x) for the cumulative P(X ≤ x).

Binomial models a fixed number of discrete trials with two outcomes; normal models continuous data across an infinite range. At HL, when n is large and p isn't too close to 0 or 1, you can approximate a binomial using the normal distribution with a continuity correction — examiners test this link directly.

Poisson models the number of events in a fixed interval of time or space using one parameter, the mean λ, while binomial needs a fixed number of trials n and a probability p. HL students sometimes approximate binomial with Poisson when n is large and p is small — the formula booklet gives the exact condition.

For X ~ B(n, p), the mean is E(X) = np and the variance is Var(X) = np(1-p). Both come straight from the IB formula booklet, so you don't derive them in an exam — you just substitute n and p correctly and show the substitution as working.

Use binompdf(n, p, x) for an exact probability P(X = x), and binomcdf(n, p, x) for the cumulative P(X ≤ x). For 'at least' or 'greater than' questions, subtract from 1, since most calculators don't have a direct inequality function built in.

The three I see most often in mock scripts: using binompdf when the question needs binomcdf (or vice versa), assuming binomial applies without checking independence, and rounding p too early in a multi-step problem. Any one of these can lose the whole mark on a question worth two or three marks.

The binomial distribution sits in the Statistics and Probability topic for both IB Maths Analysis & Approaches and Applications & Interpretation, at SL and HL. HL adds the normal approximation link and more demanding multi-stage problems that combine binomial with conditional probability.

In AA, binomial questions appear on Paper 2 and Paper 3 (HL only), since both require a GDC for binompdf/binomcdf — it won't appear on the non-calculator Paper 1. In AI, where every paper allows the GDC, binomial can show up on either Paper 1 or Paper 2, often blended with expected value or hypothesis testing.

Yes — binomial distribution is a genuinely defensible choice for a Mathematical Exploration, for instance modelling free-throw shooting or repeated coin tosses. The trap is picking a real dataset that doesn't actually meet the independence or constant-probability conditions, which examiners flag under Criterion E, Use of Mathematics.

You don't need expensive resources for one topic — a formula booklet, a GDC, and focused practice questions on binomial probability cover it. On revisionprep.com, the Topical Worksheets and Revision Notes for Statistics and Probability bundle binomial, normal and Poisson together, which works out more cost-effective than buying separate topic guides.