Statistics & Probability

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Binomial Distribution Calculator.

Calculate a binomial probability, mean, and variance.

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Set your values

Results update as you type.

probability: 0.117188

probability

0.117188
mean: 5

mean

5.000000
variance: 2.5

variance

2.500000

Results update automatically as you type.

Use Cases

Quality Control

Determine the probability of finding a certain number of defective items in a batch, given a known defect rate.

Example: If 5% of products are defective, what's the probability of 2 defects in a sample of 20?

Survey Analysis

Calculate the likelihood of a specific number of positive responses in a survey, assuming a known response rate.

Example: With a 30% response rate, what's the probability of getting 10 responses out of 50?

Frequently Asked Questions

What is the binomial distribution?
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It's used when there are only two outcomes (success/failure) per trial.
How do I use this calculator?
Enter the number of trials (n), the number of successes (k) you're interested in, and the success probability (p) for each trial. The calculator will compute the probability of exactly k successes and provide moments like mean and variance.
What are the moments in the binomial distribution?
Moments describe the distribution's shape. The mean is n*p, variance is n*p*(1-p), and standard deviation is the square root of variance. These help summarize the expected outcome and variability.

Tips & Common Mistakes

Tips

  • Ensure the success probability is between 0 and 1 (inclusive).
  • The number of successes (k) must be between 0 and the number of trials (n).
  • Use the mean (n*p) to get a quick sense of the expected number of successes.
  • For cumulative probabilities (e.g., at most k successes), you may need to sum individual probabilities.

Common Mistakes to Avoid

  • Entering a success probability as a percentage (e.g., 50 instead of 0.5).
  • Setting k greater than n, which is impossible.
  • Assuming the trials are not independent when they should be for a binomial distribution.

Last updated: August 13, 2026