Binomial distribution — mean, variance and unknown parameters (IB Math AA HL)
Updated October 2026
Quick answer: Use X ~ B(n, p) when there are n independent trials, each with the same probability p of success, and X counts the successes. Then P(X = r) = ⁿCᵣ pʳ(1 − p)ⁿ⁻ʳ, E(X) = np and Var(X) = np(1 − p). On the GDC, binomial PD gives P(X = r) and binomial CD gives P(X ≤ r).
B(n, p) with mean np and variance np(1-p). HL AA combines the binomial with conditional probability, or asks you to find n or p from a given probability.
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Check the conditions before you use B(n, p): a fixed number of trials, two outcomes, a constant probability and independent trials.
To find n, write the condition as an inequality, for example 1 − (1 − p)ⁿ > 0.9, then use logarithms or a table of values on the GDC. Round n up to the next integer.
To find p from the mean and variance: np(1 − p) ÷ np = 1 − p.
Conditional questions: P(X = 1 | X ≥ 1) = P(X = 1) ÷ P(X ≥ 1), since X = 1 is already inside X ≥ 1.
"At least" questions: P(X ≥ r) = 1 − P(X ≤ r − 1). Check whether the inequality is strict before you use the cumulative function.
When a question changes the number of trials, define a new random variable, Y ~ B(m, p), rather than reusing X.
Worked example
The probability that a component is faulty is 0.04, independently of other components. (a) Find the least number of components that must be tested so that the probability of finding at least one faulty component is greater than 0.9. (b) A batch of 20 components contains at least one faulty component. Find the probability that it contains exactly one.
(a) Let X ~ B(n, 0.04). P(X ≥ 1) = 1 − 0.96ⁿ > 0.9, so 0.96ⁿ must be less than 0.1.
Taking logs, n ln 0.96 is less than ln 0.1. Dividing by ln 0.96, which is negative, turns the inequality round: n > ln 0.1 ÷ ln 0.96 ≈ 56.4, so n = 57.
Check: 1 − 0.96⁵⁶ ≈ 0.898 and 1 − 0.96⁵⁷ ≈ 0.902.
(b) Let Y ~ B(20, 0.04). P(Y = 1) = 20 × 0.04 × 0.96¹⁹ ≈ 0.3683 and P(Y ≥ 1) = 1 − 0.96²⁰ ≈ 0.5580.
A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:
Write the condition as an inequality in n, usually with the complement, such as 1 − (1 − p)ⁿ > 0.9. Solve it with logarithms or a table of values on the GDC, and take the next whole number up, checking the values either side.
Where can I practise binomial distribution questions?
On the Advanced Random Variables questions page: 52 exam-style HL AA questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for binomial distribution?
In HL 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.