Binomial Distribution
PMF and CDF for n independent trials
About this calculator
The Binomial Distribution Calculator finds the probability of k successes in n independent trials, each succeeding with probability p. It plots the full probability mass function (PMF) and reports the exact probability P(X = k), the cumulative P(X ≤ k), the upper tail P(X ≥ k), and the range P(a ≤ X ≤ b), along with the mean np, variance np(1−p), and standard deviation. Useful for coin flips, quality control, polling, A/B testing, and probability homework.
How to use the Binomial Distribution 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
- 10 fair coin flips, exactly 5 heads: P(X = 5) = 0.2461
- 10 fair coin flips, at most 5 heads: P(X ≤ 5) = 0.6230
- 20 trials at p = 0.3, at least 10 successes: P(X ≥ 10) = 0.0480
- Between 4 and 6 heads in 10 fair flips: P(4 ≤ X ≤ 6) = 0.6563
- n = 100, p = 0.5: mean = 50, σ = 5
Frequently asked questions
How does the Binomial Distribution 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 Distribution 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.