P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}Variables
- n: number of trials
- k: number of successes
- p: success probability per trial
- X: success count
How to use this formula
Finds the probability of exactly k successes in n independent Bernoulli trials.
Important notes
- Trials must be independent and use the same success probability.
- k must be an integer from 0 through n.
Quick example
For 3 fair coin flips, the probability of exactly 2 heads is 3/8.
Applicability, worked calculation, and verification
Domain and applicability
Apply Binomial Probability Formula only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- Trials must be independent and use the same success probability.
- The event model, conditioning information, independence assumptions, and probability range must match the problem.
Do not use this formula when
- Do not use Binomial Probability Formula until the sample, event, distribution, independence assumptions, and parameter convention match the problem.
Boundary and special cases
- For Binomial Probability Formula, check zero, negative, and extreme input values before relying on the result.
- When using Binomial Probability Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form P(X=k)=\binom{n}{k}p^k(1-p)^{n-k} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For three fair coin flips, the probability of exactly two heads is C(3,2)(0.5)²(0.5)=3/8=0.375.
- Set n=3, k=2, and p=0.5.
- Compute C(3,2)=3 and p²(1-p)=0.125.
- Multiply 3×0.125 to obtain 0.375.
Independent verification
The three equally likely outcomes HHT, HTH, and THH out of eight total outcomes confirm 3/8.
Common mistakes
- Do not substitute sample and population quantities interchangeably in Binomial Probability Formula; map every symbol to its definition first.
- Do not assume events are independent or mutually exclusive unless the problem states or proves that condition.
Continue the workflow
Use Binomial Probability Formula in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Binomial Probability Formula.
Verified against: OpenStax Introductory Statistics
Automated quality check: Passed the core formula indexing gate.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Binomial Probability Formula used for?
Finds the probability of exactly k successes in n independent Bernoulli trials.
Can I copy this formula as LaTeX?
Yes. Copy P(X=k)=\binom{n}{k}p^k(1-p)^{n-k} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.