Binomial Distribution Calculator
Set the number of trials and the chance of success to get exact, at-most and at-least probabilities with the whole distribution drawn out.
Binomial formula
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
Cumulative probabilities add these terms: P(X ≤ k) sums from 0 to k. The calculator works in logarithms so that large n (thousands of trials) doesn’t overflow.
Worked example
A multiple-choice quiz has 10 questions with 4 options each. Guessing blindly, the chance of getting at least 6 right is P(X ≥ 6) with n = 10, p = 0.25, which is about 0.0197, roughly 1 in 50. For the chance of two particular events rather than a count, see the probability calculator.
Frequently asked questions
When does a problem follow a binomial distribution?
When there is a fixed number of trials n, each trial has two outcomes (success or failure), the probability of success p is the same every time, and trials are independent. Counting heads in 10 coin flips, or defective items in a batch of 50 with a known defect rate, fits.
What is the binomial probability formula?
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, where C(n, k) = n! ÷ (k!(n − k)!) counts the ways to arrange k successes among n trials.
What is the difference between “at most” and “at least”?
“At most k” means k or fewer: P(X ≤ k). “At least k” means k or more: P(X ≥ k) = 1 − P(X ≤ k − 1). “More than k” excludes k itself.
What are the mean and standard deviation of a binomial distribution?
Mean = np and standard deviation = √(np(1 − p)). With n = 100 and p = 0.3, expect 30 successes, give or take about 4.6.
When can I use the normal approximation?
A common rule is when both np and n(1 − p) are at least 10. Then X is roughly normal with mean np and SD √(np(1−p)); use a continuity correction of ±0.5. This calculator computes exact binomial probabilities, so you don’t need the approximation.