Binomial Distribution Calculator
Calculate binomial probabilities, expected value, and variance for discrete trials.
Table of Contents
How to Use
- Enter the number of independent trials, the probability of success on each trial, and the number of successes you want to evaluate.
- Click calculate to view the exact probability, cumulative probabilities, and binomial summary statistics.
- Review the distribution table to see how likelihood is allocated across different counts of successes.
What is the Binomial Distribution?
The binomial distribution models the number of successes in a fixed number of independent trials when each trial has the same probability of success. Typical examples include coin flips, pass/fail quality control tests, and customer conversions.
- n = number of trials
- k = number of observed successes
- p = probability of success on each trial
- q = 1 − p = probability of failure
Probability Mass Function
The probability of observing exactly k successes is given by: P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) is the binomial coefficient representing the number of unique arrangements.
The expected value is n × p and the variance is n × p × (1 − p), providing quick insight into the center and spread of the distribution.
Frequently Asked Questions
- When should I use the binomial distribution?
- Use the binomial distribution when you have independent trials, a fixed number of attempts, only two outcomes (success or failure), and a constant probability of success on each trial.
- Why do probabilities sometimes not sum to exactly 1?
- Floating point rounding can introduce very small errors. This calculator clamps results to stay within [0, 1], so minor discrepancies are due to numerical precision.
- How can I analyze more than 21 outcome rows?
- For large n the table only shows the first 21 rows to keep the interface readable. Export the probabilities or reduce n to view the full distribution.
Related Calculators
statistics
5 Number Summary Calculator
statistics
Absolute Deviation Calculator