Binomial Probability Calculator
Find the probability of k successes in n independent trials.
Results update as you type.
About this calculator
The binomial probability calculator uses the binomial distribution to model the number of successes in a fixed number of independent trials, each with the same success probability — the classic "how many heads in ten coin flips" problem. It returns the probability of exactly k successes, P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ, along with the cumulative probabilities of at most k and at least k successes.
The term C(n, k) is the number of ways to choose which k of the n trials succeed, pᵏ is the chance those k succeed and (1 − p)ⁿ⁻ᵏ the chance the rest fail. It also reports the distribution’s mean (n·p), variance (n·p·(1 − p)) and standard deviation, and shows the full probability distribution as a table and a chart.
Worked example: for n = 10 flips of a fair coin (p = 50%) the chance of exactly k = 3 heads is C(10, 3) × 0.5³ × 0.5⁷ = 120 × 0.5¹⁰ ≈ 11.7%. The expected number of heads is 10 × 0.5 = 5, with a standard deviation of √(10 × 0.5 × 0.5) ≈ 1.58.
Frequently asked questions
When can I use the binomial distribution?
Use it when there is a fixed number of independent trials, each trial has just two outcomes (success or failure), and the success probability is the same every time — like flipping a coin 10 times or counting defective items in a batch.
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the chance of exactly k successes, while P(X ≤ k) adds up the chances of 0, 1, … up to k successes. The cumulative version answers questions like "what is the probability of at most 3 successes?"
What is "n choose k"?
C(n, k), read "n choose k", is the number of different ways to pick which k of the n trials are the successes. For 10 trials and 3 successes there are C(10, 3) = 120 such combinations, and each contributes to the total probability.
What are the mean and standard deviation of a binomial distribution?
The mean (expected number of successes) is n·p and the variance is n·p·(1 − p), so the standard deviation is √(n·p·(1 − p)). For 10 fair coin flips the mean is 5 and the standard deviation about 1.58.
When can I approximate the binomial with a normal distribution?
When n is large and both n·p and n·(1 − p) are at least about 10, the binomial is well approximated by a normal distribution with the same mean and standard deviation. This is the basis of many large-sample proportion tests.
How do I find the probability of at least k successes?
P(X ≥ k) sums the probabilities of k, k+1, … up to n successes. It equals 1 − P(X ≤ k − 1), and this binomial probability calculator reports it directly alongside the "at most" value.
API — use this calculator from code
Call this calculator as a free JSON endpoint — no key required. Send the field values below as query parameters or JSON. Read the full API docs →
Endpoint
GET https://calculator.free/api/v1/binomial-probability/curl
curl "https://calculator.free/api/v1/binomial-probability/?n=10&k=3&p=50"JavaScript fetch()
const r = await fetch(
"https://calculator.free/api/v1/binomial-probability/?" + new URLSearchParams({
"n": "10",
"k": "3",
"p": "50"
}));
const data = await r.json();
console.log(data.results);Results are estimates for general guidance only, not financial, medical or tax advice.