Binomial Distribution Calculator

How it works

1. 1Enter n (number of independent trials), k (desired number of successes), and p (probability of success on each trial).
2. 2The calculator applies the binomial PMF C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ to find the exact probability of k successes.
3. 3The bar chart displays the full distribution P(X = k) for every k from 0 to n, so you can see its shape at a glance.

Use cases

• Quality control: estimate the probability of a specific number of defective items in a batch of n products.
• Clinical trials: determine the likelihood of exactly k patients responding to a treatment out of n enrolled.
• Games and sports: compute the chance of winning exactly k out of n rounds when each has the same win probability.

Frequently asked questions

What is the binomial distribution?

It models the number of successes in n independent Bernoulli trials, each with the same probability p of success. It is fully described by two parameters: n (trials) and p (success probability).

What is the difference between PMF and CDF?

The PMF gives the probability of exactly k successes, P(X = k). The CDF gives the cumulative probability P(X ≤ k), summing the PMF from 0 up to k. This calculator shows the PMF for each k.

Why must n be between 1 and 50?

The calculator clamps n to 50 to keep the chart readable and computation fast. For larger n, the normal approximation (mean ≈ np, sd ≈ √(np(1−p))) is accurate when np ≥ 5 and n(1−p) ≥ 5.