When should I use a binomial model?

Use it when each trial has only two outcomes, the trials are independent, and the success probability stays constant from trial to trial.

Can I enter p as a percent or a decimal?

Yes. This calculator accepts forms like 0.35, 35, or 35%. In the summary row it restates the normalized model value so you can confirm what was used.

What do exactly, at most, at least, and between mean?

Exactly uses one success count k. At most sums from 0 through k. At least sums from k through n. Between sums from k through k2, inclusive.

How is this different from statistics or permutations?

Statistics summarizes observed data after the fact. Permutations and combinations count arrangements. This calculator models the probability of success counts before or during repeated trials.

Binomial Probability Calculator

Choose exactly, at most, at least, or between successes in repeated yes/no trials and get the matching binomial probability instantly.

How It Works

Formula

P(X = k) = C(n,k)p^k(1-p)^{n-k}

C(n,k) = \frac{n!}{k!(n-k)!}

P(X \le k) = \sum_{i=0}^{k} C(n,i)p^i(1-p)^{n-i}

P(X \ge k) = \sum_{i=k}^{n} C(n,i)p^i(1-p)^{n-i}

P(a \le X \le b) = \sum_{i=a}^{b} C(n,i)p^i(1-p)^{n-i}

\mu = np, \quad \sigma = \sqrt{np(1-p)}

This calculator models $X$ as a binomial random variable. Exact mode uses $P(X = k) = C(n,k)p^k(1-p)^{n-k}$ . The at-most, at-least, and between modes add the exact probabilities over the matching range of success counts. The result panel also reports $np$ and $\sqrt{np(1-p)}$ so you can compare the requested event to the distribution’s center and spread.

Assumptions: independent trials and a constant success probability.
Probability input: 0.35, 35, and 35% all normalize to the same modeled probability.
Numerical stability: the calculator evaluates binomial terms from log-gamma identities instead of expanding factorials directly, which avoids the overflow problems of naive n! arithmetic.