Binomial distribution formula: P(X = r) = nCr pʳ(1 − p)ⁿ⁻ʳ
The binomial distribution formula is P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ.
What each letter means
- \(n\) the number of trials
- \(p\) the probability of success in one trial
- \(r\) the number of successes
- \(\binom{n}{r}\) nCr, the number of orders the successes can come in
When to use it
When there is a fixed number of independent trials, each with the same two outcomes and the same probability of success.
Worked example
\(X\sim B(8,\,0.3)\). Find \(P(X=2)\).
- \(P(X=2)=\binom82(0.3)^2(0.7)^6=28\times0.09\times0.7^6\)
Answer: \(P(X=2)=0.296\) (3 s.f.)
Common mistake
Leaving out nCr. pʳ(1 − p)ⁿ⁻ʳ is the probability of one particular order only.
On your course
| Course | In the exam |
|---|---|
| AA SL | Not in the booklet: learn it |
| AI SL | Not in the booklet: learn it |
| AA HL | Not in the booklet: learn it |
| AI HL | Not in the booklet: learn it |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
Questions
What is the binomial distribution formula?
The binomial distribution formula is P(X = r) = nCr × pʳ × (1 − p)ⁿ⁻ʳ. n: the number of trials; p: the probability of success in one trial; r: the number of successes; nCr: nCr, the number of orders the successes can come in.
Is the binomial distribution given in the exam?
AA SL: not in the booklet: learn it. AI SL: not in the booklet: learn it. AA HL: not in the booklet: learn it. AI HL: not in the booklet: learn it. This comes from our own IB Maths formula sheets; your teacher has the official booklet.
What are the mean and variance of a binomial distribution?
For X ~ B(n, p): E(X) = np and Var(X) = np(1 − p).
Our own wording, examples and card, checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.