Math · Statistics
Binomial probability calculator.
Enter the number of trials, the chance of success on each trial and the number of successes you want. Exact, cumulative and between-two-numbers answers in one go.
With n = 10 trials and p = 0.5, the chance of exactly 3 successes is 0.1172 (120 / 1024). The chance of 3 or fewer is 0.1719, of 3 or more is 0.9453, and of between 3 and 6 successes is 0.7734. The mean is np = 5 and the standard deviation is √2.5 = 1.5811.
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The formula
The binomial distribution gives the probability of x successes in n trials when each trial has two outcomes and the success probability p is fixed for all trials: P(x) = C(n, x) px (1 − p)n−x, where C(n, x) = n! / (x! (n − x)!). The cumulative probability F(x) adds P(0) through P(x). The mean is np and the standard deviation is √(np(1 − p)). At least k means 1 − P(X ≤ k − 1), and between a and b (inclusive) means F(b) − F(a − 1); the calculator adds the probabilities directly so very small tails keep their precision.
Worked example
n = 10 and p = 0.5: P(3) = C(10, 3) × 0.510 = 120 / 1024 = 0.1172. P(X ≤ 3) = (1 + 10 + 45 + 120) / 1024 = 176 / 1024 = 0.1719. P(X ≥ 3) = 1 − (1 + 10 + 45) / 1024 = 968 / 1024 = 0.9453. P(3 ≤ X ≤ 6) = (120 + 210 + 252 + 210) / 1024 = 792 / 1024 = 0.7734. Mean = 10 × 0.5 = 5; standard deviation = √(10 × 0.5 × 0.5) = 1.5811.
Limits
Use it only when the trials are independent and p really is the same every time. Sampling without replacement from a small group breaks that. The calculator accepts n up to 100,000.
Source, opened 8 October 2026: NIST/SEMATECH e-Handbook, 1.3.6.6.18 Binomial Distribution (probability mass function, cumulative function, mean np, standard deviation √(np(1 − p)), and the assumption that p is fixed for all trials).
Frequently asked questions
How do you calculate a binomial probability?
Use P(x) = C(n, x) x p^x x (1 - p)^(n - x). For n = 10, p = 0.5 and x = 3, C(10, 3) = 120, so P(3) = 120 / 1024 = 0.1172.
How do I find the probability between two numbers?
Add the exact probabilities from the lower number to the upper number, inclusive. Enter the lower number as k and the upper number in the between box. For 3 to 6 successes in 10 trials at p = 0.5 the answer is 0.7734.
What is the difference between "at most" and "fewer than"?
At most 3 means 0, 1, 2 or 3 successes, P(X <= 3). Fewer than 3 means 0, 1 or 2, P(X < 3). The calculator shows both so you can pick the wording in your question.
What are the mean and standard deviation of a binomial?
The mean is n x p and the standard deviation is the square root of n x p x (1 - p). For n = 10 and p = 0.5 those are 5 and 1.5811.
When can I not use the binomial distribution?
When trials are not independent, when p changes from trial to trial, or when there are more than two outcomes per trial. The binomial assumes p is fixed for all trials.
Can I enter p as a percent?
Yes. Typing 50% is read as 0.5.
Learning aid, not for decisions
Educational calculator. Results assume the conditions listed on this page, which real data often only roughly meet. Do not use it for clinical, regulatory, legal or financial decisions.