Binomial Pmf Calculator
Binomial pmf calculator works out P(X = k) and P(X ≤ k) from Number of trials (n), Number of successes (k) and Probability of success on one trial (p). The default figures on this page give P(X = k) of 0.3125; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
4 steps
- C(5, 2) = 10
- P(X = 2) = C(n,k) · p^k · (1−p)^(n−k) = 10 × 0.5^2 × 0.5^3 = 0.3125
- Mean = np = 5 × 0.5 = 2.5
- Variance = np(1−p) = 1.25, sd = 1.1180339887
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
2 sources
What the binomial pmf calculator calculates
Computes binomial probabilities — exactly k, at most k, at least k or a range of successes in n independent trials — with the distribution’s mean, variance and standard deviation.
How to use this binomial pmf calculator
Enter Number of trials (n), Number of successes (k) and Probability of success on one trial (p). P(X = k) updates as you type. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind P(X = k), P(X ≤ k) and P(X < k).
The method follows formula R3954 and NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.18 "Binomial Distribution".
What you can compare
Change one input at a time to see how it affects P(X = k), P(X ≤ k) and P(X < k). The inputs you can vary are Number of trials (n), Number of successes (k) and Probability of success on one trial (p).
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
How to calculate binomial PMF?
Computes binomial probabilities — exactly k, at most k, at least k or a range of successes in n independent trials — with the distribution’s mean, variance and standard deviation. Open “How this is worked out” beside the result to see each step for your own figures.
What is PMF in binomial distribution?
Computes binomial probabilities — exactly k, at most k, at least k or a range of successes in n independent trials — with the distribution’s mean, variance and standard deviation. On this page it is computed from Number of trials (n), Number of successes (k) and Probability of success on one trial (p) and shown as P(X = k), P(X ≤ k) and P(X < k).
How do I calculate the binomial distribution using a calculator?
It needs Number of trials (n), Number of successes (k) and Probability of success on one trial (p). Nothing else is required, and the figures stay in your browser.
What information do I need for the binomial pmf calculator?
You need Number of trials (n), Number of successes (k) and Probability of success on one trial (p). The page opens with default figures that give P(X = k) of 0.3125, so you can see the scale before entering your own.
What does this binomial pmf calculator show?
It shows P(X = k), P(X ≤ k), P(X < k), P(X ≥ k), P(X > k), P(k ≤ X ≤ k₂), C(n, k) and Mean (np), with the working and any warnings for your entries.
Can I check how the binomial pmf calculator result was calculated?
Yes. The “How this is worked out” section shows the calculation steps, and the sources section identifies the references used for the method.