Binomial Probability Calculator
The binomial probability calculator computes the probability of getting exactly k successes in n independent trials, where each trial has a fixed probability p of success. The formula is P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where C(n,k) is the binomial coefficient (combinations). For example, if you flip a fair coin 10 times, the probability of getting exactly 5 heads is about 24.6 percent. This calculator also shows the cumulative probability P(X <= k), which is the probability of getting k or fewer successes. A visualization shows where your parameters fall on the probability distribution.
Formula
P(X = k) = C(n,k) * p^k * (1-p)^(n-k)
where C(n,k) = n! / (k! * (n-k)!)
Distribution parameters
Binomial probability calculator: frequently asked questions
What is binomial probability?
Binomial probability describes the probability of getting exactly k successes in n independent trials, where each trial has probability p of success. The formula is P(X=k) = C(n,k) * p^k * (1-p)^(n-k).
What is cumulative binomial probability?
Cumulative binomial probability P(X <= k) is the sum of probabilities P(X=0) + P(X=1) + ... + P(X=k). It answers questions like: what is the probability of at most k successes?
What does the bell curve show?
For larger n, the binomial distribution approaches a normal (bell) curve. The mean is n*p and the standard deviation is sqrt(n*p*(1-p)). This visualization shows where your parameters fall on this distribution.
When is binomial probability used?
Binomial probability is used for coin flips, quality control testing, survey questions with yes/no answers, or any situation with a fixed number of independent trials and two possible outcomes per trial.
What if p is not 0.5?
If p is not 0.5, the distribution becomes skewed. If p < 0.5, it skews right (towards more failures). If p > 0.5, it skews left (towards more successes).
Official sources
- Wolfram MathWorld: Binomial Distribution.
- Wikipedia: Binomial Distribution.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.