A familiar situation
Estimate the chance of a chosen number of successes across repeated independent trials.
How to use this calculator
- Enter number of trials, success count or threshold, success probability and the remaining inputs.
- Select Calculate to see the answer and its breakdown.
- Keep a scenario, change an input and compare the results.
Use consistent units and check the assumptions below before applying the result.
How probability is calculated
For X~Binomial(n,p), P(X=k)=C(n,k)p^k(1−p)^(n−k). Cumulative options sum the relevant outcomes from 0 through n. “At most k” includes zero through k; “at least k” includes k through n. The model describes a fixed number of independent trials with one shared success chance, such as repeated draws with replacement or repeated fair coin flips. With 5 fair coin flips, exactly 2 heads has probability C(5,2)/2⁵=10/32=31.25%.
P(X=k)=C(n,k)p^k(1−p)^(n−k), X~Binomial(n,p).With 5 fair coin flips, exactly 2 heads has probability C(5,2)/2⁵=10/32=31.25%.
Assumptions & useful context
The result depends on the values you enter and the method described above. Displayed numbers may be rounded; calculations use the unrounded values. Check that this model matches your situation before relying on it.
Frequently asked questions
What assumptions does the binomial model make?
A fixed trial count, independent trials, and a constant success probability.
Does “at least” include the chosen count?
Yes, it sums k through n.