Binomial Probability Calculator
Find the probability of exactly, at most, or at least k successes in n independent trials, each with the same success probability p.
How it works
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ · where C(n, k) = n! ÷ [k!·(n − k)!]
The binomial distribution models the number of successes in a fixed number of independent yes/no trials that each succeed with the same probability p. The formula multiplies three things: how many distinct ways k successes can occur among n trials, the probability of k successes, and the probability of the remaining n − k failures. Cumulative "at most" and "at least" probabilities add up the individual terms.
Worked example
Flip a fair coin 10 times (n = 10, p = 0.5) and ask for exactly 4 heads. C(10, 4) = 210, so P(X = 4) = 210 · 0.5⁴ · 0.5⁶ = 210 · 0.0009766 ≈ 0.205 — about a 20.5% chance.
Sources: Binomial distribution — Wikipedia
Frequently asked questions
When can I use the binomial distribution?
You need a fixed number of trials, only two outcomes per trial (success or failure), the same success probability on every trial, and independent trials. Coin flips, pass/fail tests and defect counts on a line all fit; drawing cards without replacement does not, because the probability changes each draw.
What is the difference between "at most" and "at least" k?
P(X ≤ k) adds the probabilities of 0, 1, … up to k successes, while P(X ≥ k) adds k, k + 1, … up to n. They overlap only at exactly k, which is why P(X ≥ k) = 1 − P(X ≤ k) + P(X = k).
How large can n be here?
This calculator works in logarithms, so it stays accurate for n up to 100,000 without overflowing on the factorials. For very large n the normal approximation with mean n·p and variance n·p·(1 − p) gives nearly the same answer.
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For education — check the assumptions (sample size, independence, distribution) before using these results in real decisions.