Calculator
What the Binomial Probability Calculator does
Binomial Probability Calculator: calculates P(X = k) and Percent from Trials n, Successes k and Success Probability p. Example: trials n 10, successes k 3 and success probability p 0.5 gives P(X = k) 0.1172.
From Trials n, Successes k and Success Probability p, this tool derives P(X = k) and Percent. The result recalculates on every keystroke without a button press or a page reload.
It is mostly used by people who want the working shown. Checking a result a second way is the cheapest way to catch an arithmetic slip.
Everything happens inside this page. Nothing you type is sent to a server, and there is nothing for us to store.
The method, a field-by-field reference and a worked example built from the defaults are all on this page.
What do the Binomial Probability Calculator fields mean?
The Binomial Probability Calculator uses 3 inputs. The form opens pre-filled, so the fastest way in is usually to overwrite the one field you actually care about.
| Field | What to enter | Default |
|---|---|---|
| Trials n | required | 10 |
| Successes k | required | 3 |
| Success Probability p | accepts 0 to 1 | 0.5 |
How does the Binomial Probability Calculator work?
Each output is derived from the inputs above in a single pass; there is no hidden state carried between runs, so the same inputs always give the same calculation.
Nothing is computed until the inputs pass their checks, so you will not get a confident answer built on a typo.
Worked example
These are the values the Binomial Probability Calculator loads by default, and the result it produces from them. These are the tool’s own default values and the result they produce, not an invented illustration.
| Trials n | 10 |
|---|---|
| Successes k | 3 |
| Success Probability p | 0.5 |
| P(X = k) | 0.1172 |
|---|---|
| Percent | 11.7188 |
| What this means | P(X=k) ≈ 0.1171875 |
Beyond the formula: binomial probability
The binomial formula only applies when trials are independent and each has exactly the same success probability every time. Sampling without replacement from a small population, where each draw slightly changes the odds for the next, technically violates this and calls for the hypergeometric distribution instead.
Common mistakes and sanity checks
- "Exactly k successes," what this calculator computes, is a different (usually much smaller) number than "at least k successes" or "at most k successes." Those require summing the binomial formula over a range of k values, not evaluating it once.
- The combination term C(n,k) inside the formula counts how many different orderings of successes and failures achieve exactly k successes: the same combinatorics as the standalone combination calculator, just embedded inside a larger formula.
- As n grows large, the binomial distribution increasingly resembles a normal distribution centred at n×p: a fact used as a shortcut approximation before computers made exact binomial calculation routine.
How to use it
- Fill in Trials n.
- Type Successes k.
- Enter Success Probability p (0 to 1).
- Type a value and the answer is already there by the time you stop.
- The panel reports P(X = k), Percent and What this means.
- Both Copy Link and Print capture the current state, so pick whichever suits how you file things.
What this tool does not do
- Displayed values are rounded; the underlying calculation keeps full precision.
- Very large or very small numbers hit JavaScript floating-point limits (about 15 significant digits).
- It applies one method. If your situation calls for a different convention, the answer will not match.
Frequently asked questions
Just Trials n, Successes k and Success Probability p. A starting value is in place for each, which makes it easy to isolate the effect of a single variable.
Success Probability p accepts 0 to 1. If a value falls outside, the tool says which field is at fault instead of guessing.
With the values loaded when the page opens, P(X = k) comes out as 0.1172. It recalculates the moment an input changes, which makes comparing scenarios quick.
It never leaves the tab. The page does the work itself rather than asking a server to.
It costs nothing, and there is no limit on repeat use.
The output is deterministic. The same inputs always give the same answer, and the method is documented above so you can check it.