How to Use the Binomial Distribution Calculator
The binomial distribution describes the probability of getting exactly k successes out of n independent trials, where each trial has the same probability p of success. It is widely used in statistics, quality control, biology, finance, and everyday probability questions like coin tosses or dice rolls.
Enter the number of trials (n), the probability of success per trial as a percentage, and the target number of successes (k). The calculator computes P(X=k) — the exact probability of k successes, P(X≤k) — the cumulative probability of at most k successes, and P(X≥k) — the probability of at least k successes. The expected value E(X)=n×p shows the average expected successes.
This tool supports up to 1,000 trials. For very large n values, the binomial distribution can be approximated by the normal distribution. Each trial must be independent and have the same probability of success for the binomial model to apply.
Frequently Asked Questions
Use it when trials are independent, each has exactly two outcomes (success/failure), and the success probability is constant across trials. Examples include coin flips, pass/fail tests, and defect rates.
The calculator uses exact computation via log-factorials for precision. For n above 1,000, the normal approximation (μ=np, σ=√(np(1-p))) provides a very accurate estimate.
P(X≥k) = 1 − P(X≤k−1). It equals one minus the cumulative probability of k−1 or fewer successes.