Binary Distribution Calculator
Binomial Distribution 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 Distribution 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 Distribution Calculator
Enter Number of trials (n), Number of successes (k), Probability of success on one trial (p) and Upper k (for a range). P(X = k) updates as you type, and “Use a worked example” loads checked figures you can adjust. 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".
Research R3954-GT-1: 2 heads in 5 fair coin tosses example
With Number of trials (n): 5, Number of successes (k): 2, Probability of success on one trial (p): 0.5, the calculator returns P(X = k): 0.3125, C(n, k): 10. This example is checked against an independent source in the test suite.
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), Probability of success on one trial (p), Upper k (for a range) and Question asked.
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 find binomial distribution on Casio FX-CG50?
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.
How to press binomial in calculator?
It needs Number of trials (n), Number of successes (k), Probability of success on one trial (p), Upper k (for a range) and Question asked. Nothing else is required, and the figures stay in your browser.
How do you calculate a binomial distribution?
The result gives P(X = k), P(X ≤ k), P(X < k), P(X ≥ k), P(X > k) and P(k ≤ X ≤ k₂), each labelled with its unit so the figure can be read on its own.
How do I use the binomial theorem on a calculator?
The page opens with worked default figures: P(X = k) of 0.3125. Replace them with your own and every figure updates as you type.
What information do I need for the Binomial Distribution Calculator?
You need Number of trials (n), Number of successes (k), Probability of success on one trial (p), Upper k (for a range) and Question asked. 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 Distribution 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.