Binomial Distribution Calculator (with Graph)
Find binomial probabilities for exactly, at most, at least or between k successes.
When the binomial distribution applies
The binomial distribution counts the number of successes in a fixed number of independent trials, when every trial has the same probability of success. Four conditions must hold: a fixed number of trials n, two possible outcomes for each trial (success or failure), the same probability p each time, and independent trials. Examples are coin flips, the number of defective items in a batch, or the number of people in a sample who prefer a product.
The formula
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ, where C(n, k) is the number of ways to choose which k of the n trials succeed. The mean is n × p, the variance is n × p × (1 − p), and the standard deviation is its square root.
Worked examples
| Question | n | p | Answer |
|---|---|---|---|
| Exactly 5 heads in 10 fair coin flips | 10 | 0.5 | 0.2461 |
| At most 2 defective items in a batch of 20 | 20 | 0.1 | 0.6769 |
| Between 4 and 8 successes (inclusive) | 20 | 0.3 | 0.7796 |
| At least 7 heads in 10 flips | 10 | 0.5 | 0.1719 |
The graph under the result highlights the event you chose, and has Download PNG and Download SVG buttons for reports and slides. A table lists P(X = k), P(X ≤ k) and P(X ≥ k) for the values around the mean.
Tips
- “At least one success” is easiest as 1 − P(X = 0).
- For large n, the binomial is close to a normal distribution with mean np and standard deviation √(np(1 − p)), when both np and n(1 − p) are at least 10. The normal distribution calculator can then be used.
- If you sample without replacement from a small population, the trials are not independent, and the hypergeometric distribution is the right model.
- If you count events in a period of time and there is no fixed number of trials, use the Poisson distribution.
How to cite this page
CalcPulse. (2026). Binomial Distribution Calculator. https://calcpulse.com/binomial-distribution-calculator/
Frequently asked questions
What is a binomial distribution?
The distribution of the number of successes in a fixed number of independent trials, each with the same probability of success.
How do I calculate a binomial probability?
Use P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ. For 'at most' or 'at least', add the probabilities of each value, as the calculator does.
How do I find 'at least one' success?
Calculate 1 − P(X = 0), which equals 1 − (1 − p)ⁿ.
What are the mean and variance of a binomial distribution?
The mean is n × p and the variance is n × p × (1 − p).
When can I use the normal approximation?
When n × p and n × (1 − p) are both at least about 10. Add a continuity correction of 0.5 for the best accuracy.
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