Binomial Distribution (AI HL)
A binomial random variable counts the number of successes across a fixed number of independent trials, each with the same success probability. This page covers the mean and variance formulas, how to spot when the model applies, and the GDC routines you'll use to find binomial probabilities - with worked examples and the mistakes that lose the most marks. It's part of the broader Probability & Distributions topic.
16 questions on this sub-topic.
Mean and variance
Covered under IB syllabus reference SL4.8: the binomial distribution, its mean and variance, and the situations where it's an appropriate model, found using technology. Neither formula is in the formula booklet, so both are worth memorising.
Mean
\(X\sim B(n,p)\): \(E(X)=np\)
Not in the formula booklet. The average number of successes over \(n\) trials.
Variance
\(\text{Var}(X)=np(1-p)\)
Not in the formula booklet. Note the \((1-p)\) factor - it's easy to drop under time pressure.
Need the full syllabus wording and formula-booklet reference table? See Probability & Distributions. For calculator routines, see the parent topic's GDC guidance.
Worked examples
\(X \sim B(40, 0.3)\).
(a) Find \(E(X)\).
(b) Find \(\text{Var}(X)\).
Worked solution
(a) \(E(X) = np = 40(0.3)\) M1
\(= 12.\) A1
(b) \(\text{Var}(X) = np(1-p) = 40(0.3)(0.7)\) M1
\(= 8.4.\) A1
In a factory, 4% of items are defective. A sample of 50 is taken. Let \(X\) be the number defective, modelled as binomial.
(a) Find \(P(X = 0).\)
(b) Find \(P(X \le 3).\)
Worked solution
(a) \(P(X=0) = (0.96)^{50}\) M1
\(\approx 0.130.\) A1
(b) \(P(X \le 3) \approx 0.861.\) M1 A1
A seed germinates with probability 0.8. A gardener wants the probability of at least one germinating to exceed 0.999.
(a) Write an inequality for the number of seeds \(n\).
(b) Find the least \(n\).
Worked solution
(a) \(P(\ge 1) = 1 - (0.2)^n > 0.999 \Rightarrow (0.2)^n < 0.001.\) M1 A1 - \((0.2)^n<0.001\)
(b) \(n > \dfrac{\ln 0.001}{\ln 0.2} \approx 4.29\), so \(n\) M1
\(= 5.\) A1 A1
Common mistakes
- Assuming a variable is binomial without checking independence. The binomial model needs a fixed number of independent trials with a constant success probability - sampling without replacement from a small population breaks this and the model no longer applies exactly.
- Dropping the \((1-p)\) factor in the variance. \(\text{Var}(X)=np(1-p)\), not \(np\) - it's easy to write down the mean formula twice under exam pressure and forget the extra factor.
- Using \(P(X\le k)\) when the question asks for \(P(X
Ready to practise properly?
17 binomial-distribution questions, marked instantly like the real exam.
Quick answers
What is the mean of a binomial distribution?
If \(X \sim B(n,p)\), the mean is \(E(X) = np\).
What is the variance of a binomial distribution?
If \(X \sim B(n,p)\), the variance is \(\text{Var}(X) = np(1-p)\).