Binomial Probability Calculator
Find the probability of a number of successes in a fixed number of trials.
What the Binomial Probability Calculator does
The binomial distribution gives the probability of getting exactly k successes in n independent trials, each with the same success probability. It underpins quality control sampling, A/B testing and any repeated yes-or-no experiment.
Formula
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏMean = n × pVariance = n × p × (1 − p)
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Number of trials (n) | number | Yes | Accepts 1 or more, up to 1000. |
| Probability of success per trial | % | Yes | — |
| Number of successes (k) | number | Yes | Accepts 0 or more. |
How to use it
- Enter Number of trials (n), Probability of success per trial and Number of successes (k).
- Select Calculate.
Worked example
Flipping a fair coin 10 times — what is the chance of exactly 7 heads?
- Trials
- 10
- Probability
- 50
- Successes
- 7
C(10,7) × 0.5¹⁰ = 120 ÷ 1024 = 0.1172, about 11.72%.
Reading the result
- "P(at least k)" and "P(more than k)" differ by exactly the exact-k probability — at-least includes k, more-than excludes it, so it is worth reading the label precisely when the two are close.
- The distribution table is automatically centred and trimmed to roughly ±4 standard deviations around the mean, which captures essentially all the real probability mass for practical purposes.
Assumptions and limitations
- Every trial is assumed to share the exact same success probability and be fully independent — this excludes sampling without replacement from a small population, where the FAQ points to the hypergeometric distribution instead.
- The FAQ's normal-approximation threshold (np and n(1−p) both above roughly 5) is not checked or flagged by the calculator itself.
Common mistakes
- Summing individual probabilities by hand for an "at least" or "at most" figure and forgetting the boundary term at k — the calculator's separate "fewer than"/"more than" fields exist to sidestep exactly that off-by-one error.
Frequently asked questions
When can I use the normal approximation?
When both np and n(1−p) exceed about 5. Below that the binomial is noticeably skewed and the approximation misleads.
What if trials are not independent?
Then the binomial does not apply. Sampling without replacement from a small population needs the hypergeometric distribution instead.