Binomial Distribution (AA HL)
The binomial distribution counts the number of successes in a fixed number of independent, identical trials - passing a set of multiple-choice questions, a batch of faulty components, repeated coin flips. This sub-topic covers the probability formula, the mean and variance, and when the model is actually valid to use. It's part of the broader Probability Distributions topic.
11 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL4.8. Both formulas are in the formula booklet, so the real skill is recognising when a situation is binomial: a fixed number of trials \(n\), exactly two outcomes each time, independent trials, and a constant probability of success \(p\) throughout.
Binomial probability
\(P(X=r)=\dbinom{n}{r}p^r(1-p)^{n-r}\)
Gives the probability of exactly \(r\) successes out of \(n\) trials - on the GDC this is binompdf, or binomcdf for "at most"/"at least".
Mean & variance
\(E(X)=np,\ \ \text{Var}(X)=np(1-p)\)
Both come straight from \(n\) and \(p\) - no need to sum the full probability distribution by hand.
Need the full syllabus wording and formula-booklet reference table? See Probability Distributions. For calculator steps, see the parent topic's GDC guidance.
Worked examples
\(X\sim B(10, 0.3).\) Find \(P(X=4).\)
Worked solution
\(P(X=4) = \binom{10}{4}(0.3)^4(0.7)^6\) M1
\(= 210(0.0081)(0.117649)\) A1
\(\approx 0.200.\) A1
\(X\sim B(8, 0.25).\) Find \(P(X\ge2).\)
Worked solution
\(P(X\ge2) = 1 - P(0) - P(1).\) M1
\(P(0) = 0.75^8 \approx 0.1001\); A1 \(P(1) = 8(0.25)(0.75)^7 \approx 0.2670.\) A1
\(P(X\ge2) \approx 1 - 0.367\) M1 \(= 0.633.\) A1
A random variable \(X \sim B(6, 0.5).\) Find \(P(X = 3).\)
Worked solution
\(P(X=3) = \binom{6}{3}(0.5)^3(0.5)^3\) M1
\(= 20(0.5)^6 = \dfrac{20}{64}\) A1
\(= 0.3125.\) A1
A multiple-choice quiz has 12 questions, each with probability \(0.25\) of a correct guess.
Find the expected number correct and \(P(X=3).\)
Worked solution
\(E(X) = 12(0.25)\) M1
\(= 3.\) A1
\(P(X=3) = \binom{12}{3}(0.25)^3(0.75)^9\) M1
\(\approx 0.258.\) A1
Common mistakes
- Using binomial when the conditions don't hold. Binomial needs a fixed number of independent trials with a constant success probability - drawing without replacement, or a probability that changes between trials, rules it out.
- Forgetting probabilities must sum to 1. This is the standard way to find an unknown constant in a discrete distribution problem built from binomial probabilities - if the values you've written down don't sum to 1, something upstream is wrong.
- Mixing up pdf and cdf on the GDC. binompdf gives \(P(X=r)\) for one exact value; binomcdf gives the cumulative \(P(X\le r)\). For "at least" or "more than" you need to combine cdf with a complement, not read it off directly.
Ready to practise properly?
10 binomial-distribution questions, marked instantly like the real exam.
Quick answers
What conditions must hold for a distribution to be binomial?
A fixed number of trials \(n\), each with exactly two outcomes, independent of one another, and a constant probability of success \(p\) across every trial.
What are the mean and variance of a binomial distribution?
For \(X\sim B(n,p)\), the mean is \(E(X)=np\) and the variance is \(\text{Var}(X)=np(1-p)\).