For checking probability exercises
Binomial distribution calculator
Enter n, p and k. Get one probability, its percentage and a plain-English explanation.
Loading calculator…
After calculating, explore the steps, mean, variance, standard deviation, probability table and graph. Need a cutoff? Use the inverse CDF.
Find the explanation you need
Exactly k successes, with combinations and formula steps.
Cumulative probability, tail boundaries and inverse cutoffs.
Read the question, choose the event and check each step.
The one-count difference between ≥ and >, with examples.
How to calculate binomial probability
Enter your values.
n is the number of trials. p is the success probability for each trial, from 0 to 1. k is the success count you want to compare.
Match your question.
Choose exactly, at most, at least, less than or more than k. Confirm that independent, two-outcome trials share the same success probability.
Calculate and check.
Read the event beside your result, then inspect or copy it. Editing any input clears the previous answer so you can calculate again.
At least, at most,
or exactly?
The boundary changes the answer. At least and at most include k. More than and less than exclude it.
Worked example · separate from your result
For n = 10 and p = 0.25, fewer than 4 successes means 0, 1, 2 or 3 successes. Choose Less than k and enter k = 4.
P(X < 4) = P(X ≤ 3) ≈ 0.775875
That is about 77.5875%. “At most 4” would include 4 as well and give a different answer.
| Your wording | Event | Counts included |
|---|---|---|
| Exactly k | X = k | k only |
| At most k | X ≤ k | 0 through k |
| At least k | X ≥ k | k through n |
| Less than k | X < k | 0 through k − 1 |
| More than k | X > k | k + 1 through n |
An event with no possible counts, such as fewer than 0 successes, has probability 0.
Binomial formula, PMF and CDF
The probability mass function (PMF) gives the probability of exactly k successes in n independent trials, each with success probability p.
P(X = k) = C(n,k) pk (1−p)n−k
C(n,k) counts the ways to choose k successes from n trials. The cumulative distribution function (CDF) adds probabilities from 0 through k, giving P(X ≤ k). This calculator sums the probabilities for the one event you choose.
Read NIST’s binomial distribution reference ↗What to check before you calculate
Does the binomial model fit?
Use a fixed number of independent trials, with two outcomes and the same success probability each time. This tool does not select a distribution or solve a written question for you.
Which values are supported?
n and k must be whole numbers, with 0 ≤ n ≤ 1000 and 0 ≤ k ≤ n. Enter p as a decimal from 0 to 1. These are this version’s limits.
Why might a result use scientific notation?
A small nonzero probability may appear as 1e-1000. A value near 1 may appear as 1 minus a small probability. This keeps very small probabilities distinct from zero and near-certainty distinct from certainty.
Privacy and use
Calculations run in your browser. We don’t upload your inputs or results, save a calculation history or put your numbers in a shareable URL. The copy button writes only to your clipboard.
Cloudflare provides hosting and cookie-free visit statistics. Optional Microsoft Clarity session analytics load only after adult consent. Calculator inputs and results are masked, and our task events contain no entered numbers. Declining analytics does not affect the calculator. You can change your choice below.
BinomialCalc is a free tool operated by Vidream. Displayed probabilities are rounded. You remain responsible for choosing a suitable model and checking results before using them in a real decision. This page does not make medical, financial or quality-release decisions.