Binomial Probability Calculator
Calculate the exact probability of a specified number of successes in a fixed number of independent trials with constant success probability.
How to use this calculator
- Enter the number of trials, exact successes and success chance as a percentage.
- Check that all inputs use the same units, time period and population scope.
- Change one assumption at a time when comparing scenarios.
Formula used by binomial probability calculator calculator
P(X = k) = C(n,k) × p^k × (1 − p)^(n − k).
- P(X = k) = C(n,k) × p^k × (1 − p)^(n − k).
- The binomial model assumes independent trials, a fixed number of trials, and the same success probability on each trial. Numerical precision can be limited for extreme parameters.
- The tool validates entered values and formats results for readability; it does not verify assumptions against external records.
Worked examples
Worked example
10 independent trials, success probability 20%, exactly 3 successes.
C(10,3) × 0.2³ × 0.8⁷ ≈ 20.13%.
The probability of exactly 3 successes is about 20.13%.
Understanding the result
Calculate the exact probability of a specified number of successes in a fixed number of independent trials with constant success probability.
The binomial model assumes independent trials, a fixed number of trials, and the same success probability on each trial. Numerical precision can be limited for extreme parameters.
Important notes
The binomial model assumes independent trials, a fixed number of trials, and the same success probability on each trial. Numerical precision can be limited for extreme parameters.
Treat the result as an estimate based on the entered inputs. For regulated, engineering, employment, tax or business decisions, check applicable official rules, specifications and professional guidance.
Frequently asked questions
When is the binomial model appropriate?
Use it when each trial has two outcomes, trials are independent and the success probability is constant.
Can successes exceed trials?
No. The calculator rejects a success count greater than the number of trials.
How should I use the result?
The binomial model assumes independent trials, a fixed number of trials, and the same success probability on each trial. Numerical precision can be limited for extreme parameters.
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Related explanations
Learn the method behind the number or compare the terminology before using the calculator.
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This page provides a calculation tool plus explanatory information so you can understand what the result means and when the underlying method may not fit your situation.