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Binomial Probability Calculator

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

Modeled event
Examples

A fair coin with 10 independent flips and a target of exactly 4 heads.

Probability
20.507813 %
Probability (decimal)
0.205078125
Expected successes (np)
5
Standard deviation
1.581139
Modeled event
Exactly 4 of 10; p = 0.5 (50%)

Model only — assumes independent trials and a constant success probability. Real outcomes can differ when those assumptions do not hold.

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Examples

How It Works

Formula

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

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

P(X≤k)=∑i=0kC(n,i)pi(1−p)n−iP(X \le k) = \sum_{i=0}^{k} C(n,i)p^i(1-p)^{n-i}

P(X≥k)=∑i=knC(n,i)pi(1−p)n−iP(X \ge k) = \sum_{i=k}^{n} C(n,i)p^i(1-p)^{n-i}

P(a≤X≤b)=∑i=abC(n,i)pi(1−p)n−iP(a \le X \le b) = \sum_{i=a}^{b} C(n,i)p^i(1-p)^{n-i}

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

Variables

nn

Number of trials

pp

Probability of success on each trial

kk

Target number of successes

a,ba, b

Inclusive lower and upper bounds for a success range

XX

Random variable counting successes across n trials

μ,σ\mu, \sigma

Expected successes and standard deviation of the model

This calculator models XX as a binomial random variable. Exact mode uses P(X=k)=C(n,k)pk(1−p)n−kP(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 npnp and np(1−p)\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.

Frequently Asked Questions

01When 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.
02Can 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.
03What 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.
04How 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.

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