Exact probability
Calculates the chance of obtaining exactly a specified number of successes.
Advanced Math and Statistics
Binomial Distribution Calculator
Calculate exact, cumulative, at-least, and interval probabilities for repeated independent trials with a fixed probability of success.
- Exact and cumulative probabilities
- Mean, variance, and standard deviation
- Interactive probability distribution
Distribution inputs
Enter the number of trials, probability of success, and the probability statement to solve.
Binomial experiment
%
Success count
Quick examples
Advanced Math and Statistics
Model repeated success-or-failure experiments
The binomial distribution measures the probability of obtaining a specified number of successes in a fixed number of independent trials. Each trial must have the same probability of success and only two possible outcomes.
Cumulative probability
Finds the probability of no more than or fewer than a selected success count.
Upper-tail probability
Calculates the probability of at least or more than a selected number of successes.
Interval probability
Adds exact probabilities across an inclusive range of possible success counts.
Distribution statistics
Reports expected successes, variance, standard deviation, mode, and failure probability.
Probability visualization
Displays every possible success count with highlighted bars for the requested event.
Core formulas
Binomial probability and distribution statistics
Exact probability
Probability of exactly x successes in n independent trials.
P(X = x) = C(n,x)pˣ(1−p)ⁿ⁻ˣ
Mean and variance
The expected number of successes and the squared spread of the distribution.
μ = np and σ² = np(1−p)
Standard deviation
The typical distance of the success count from its expected value.
σ = √[np(1−p)]
How it works
Calculate binomial probability in four steps
1
Enter the trial count
Add the fixed number of independent trials in the experiment.
2
Set success probability
Enter the probability of success on each trial as a percentage.
3
Choose the event
Select exact, cumulative, at-least, greater-than, or interval probability.
4
Review the distribution
See the result, summary statistics, distribution bars, and the complete probability table.
Frequently asked questions
Binomial distribution calculator FAQs
A binomial distribution gives the probabilities for the number of successes in a fixed number of independent trials when every trial has the same success probability and only two possible outcomes.
The number of trials must be fixed, trials must be independent, each trial must have two outcomes, and the probability of success must stay constant across all trials.
Multiply the number of combinations of n trials taken x at a time by p raised to x and by one minus p raised to n minus x. The calculator performs this computation automatically.
At most x includes every success count from zero through x. At least x includes every success count from x through the total number of trials.
The mean is np and represents the expected number of successes. The standard deviation is the square root of np(1−p) and measures the typical spread around that expected value.
Yes. The probability calculation uses logarithms to reduce overflow problems for large combinations. The distribution chart automatically limits the visible range when the trial count is very large, while the requested probability remains calculated.