Binomial Distribution Calculator
Calculate binomial probabilities for n trials with success probability p.
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Binomial Distribution Calculator by TheCalcTools
How the Binomial Distribution Calculator works
Enter the number of independent trials, the chance of success on each, and the number of successes k.
The exact probability uses the binomial formula.
Cumulative rows add the probabilities of all outcomes up to or from k.
The formula
Worked example
Using the default values shown in the calculator:
| Trials (n) | 10 |
|---|---|
| Success probability (p, %) | 50 |
| Successes (k) | 3 |
- P(X = 3) = C(10,3) × 0.5³ × 0.5⁷ = 120 ÷ 1024 = 0.1172.
- P(X ≤ 3) = (1 + 10 + 45 + 120) ÷ 1024 = 0.1719.
- Mean = 5.00 and standard deviation = 1.5811.
Result:
| P(X = k) | 0.1172 |
|---|---|
| P(X ≤ k) | 0.1719 |
| P(X ≥ k) | 0.9453 |
| Mean (np) | 5.00 |
| Standard deviation | 1.5811 |
Frequently asked questions
When can I use the binomial distribution?
When there is a fixed number of independent trials, each with two outcomes and the same success probability.
What is the difference between P(X = k) and P(X ≤ k)?
The first is the chance of exactly k successes. The second adds up the chances of 0, 1, …, k successes.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.