Binomial Probability
Binomial probability and cumulative distribution: exactly, at most, at least k successes
About this calculator
The Binomial Probability Calculator finds the chance of k successes in n independent trials with success probability p, using P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ. It reports the exact probability, the cumulative tails P(X ≤ k) and P(X ≥ k), and the distribution's mean np, variance np(1 − p), and standard deviation, plotting the full probability mass function. Probabilities are computed in log space with a Lanczos log-gamma, so they stay accurate up to n = 5000. Useful for coin flips, quality control, and polling.
How to use the Binomial Probability calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- 3 heads in 10 fair coin flips → P(X = 3) ≈ 0.1172, P(X ≤ 3) ≈ 0.1719, mean 5
- n = 20 trials, p = 0.25 → P(X = 5) ≈ 0.2023, P(X ≤ 5) ≈ 0.6172
- All 6 of 6 flips heads → P = 0.015625 (1 in 64)
- 60 or more heads in 100 flips → P(X ≥ 60) ≈ 0.0284, with mean 50 and σ = 5
Frequently asked questions
How does the Binomial Probability calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Binomial Probability calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.