Quick answer: Use X ~ B(n, p) when there are n independent trials, each with the same probability p of success. P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ, E(X) = np and Var(X) = np(1 − p).
n trials, probability p of success. SL AA expects you to know both the mean (np) and variance (np(1-p)) plus how to use the GDC's binomialCDF function.
Practise Binomial distribution — B(n,p) with AI marking
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Check the four conditions in context: a fixed number of trials n, two outcomes per trial, the same probability p each time, and independent trials.
P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ. On the GDC, binomial PDF gives P(X = r) and binomial CDF gives P(X ≤ r).
Turn every inequality into a CDF before you press anything: P(X ≥ r) = 1 − P(X ≤ r − 1), and P(X > r) = 1 − P(X ≤ r). The off-by-one here is where most marks go.
E(X) = np and Var(X) = np(1 − p) are in the formula booklet; the standard deviation is the square root of the variance.
Worked example
A quiz has 12 multiple-choice questions, each with 4 options. Sam guesses every answer. Find the probability that Sam gets exactly 3 right, the probability that Sam gets at least 5 right, and the expected number right.
Each guess is right with probability 1/4, independently, so X ~ B(12, 0.25).
A calculator is allowed on AA Paper 2, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:
How do I find P(X ≥ r) for a binomial distribution?
Write it as 1 − P(X ≤ r − 1) and use binomial CDF on the GDC. For example, P(X ≥ 5) = 1 − P(X ≤ 4).
Where can I practise binomial distribution questions?
On the Discrete Random Variables questions page: 50 exam-style SL AA questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for binomial distribution?
In SL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one. The Binomial and Poisson distributions guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.