How it works
P(X=k) = n!/[k!(n−k)!] × p^k × (1−p)^(n−k). Sum these probabilities for the selected tail.
Input definitions
- Independent trials (0–1000)
- Enter the value in the labeled unit.
- Successes (k)
- Enter the value in the labeled unit.
- Success probability (0–1)
- Enter the value in the labeled unit.
- Probability
- Choose Exactly k / At most k / At least k.
A worked example
- Independent trials (0–1000)
- 10
- Successes (k)
- 5
- Success probability (0–1)
- 0.5
- Probability
- Exactly k
Assumptions & limits
Identical independent Bernoulli trials with fixed success probability; 0–1 probability input, not a percent.
Sources & review
Maintained by AnsRio. Methods and examples use automated checks; this does not imply independent professional certification.
Method updated 2026-10-17. Content checked against the stated method and examples. Our verification approach.
Common questions
What is the difference between exactly, at most and at least k?
Exactly k uses one probability mass. At most k includes all counts from zero through k. At least k includes k through the total trial count. Enter p as 0.5 for 50%.
Why can statistical results differ between tools?
Denominator, interpolation and boundary conventions can differ. Use the method and assumptions on this page to compare the same definition; displayed decimals may also be rounded.
How does the Binomial probability tool work?
P(X=k) = n!/[k!(n−k)!] × p^k × (1−p)^(n−k). Sum these probabilities for the selected tail.
What are the assumptions and limits of Binomial probability?
Identical independent Bernoulli trials with fixed success probability; 0–1 probability input, not a percent.