What is the binomial probability formula?

P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ, where n is the number of trials, p is the probability of success on each trial, and k is the exact number of successes.

How is this different from the Binomial Coefficient Calculator?

The Binomial Coefficient Calculator finds C(n,k) alone — the number of ways to choose k successes from n trials. This calculator uses C(n,k) as one factor in a full probability calculation that also accounts for how likely each success and failure combination is.

What conditions must be met to use the binomial distribution?

There must be a fixed number of trials (n), each trial must be independent, each trial has only two outcomes (success/failure), and the probability of success (p) must be the same on every trial.

What are the mean and variance of a binomial distribution?

Mean = n × p, and variance = n × p × (1 − p). For example, flipping a fair coin (p = 0.5) 100 times gives a mean of 50 successes and a variance of 25.

What does P(X ≤ k) tell me?

It's the cumulative probability of getting k or fewer successes — the sum of P(X = i) for every i from 0 to k, useful for questions like 'what's the probability of at most 3 defects in a batch of 20?'

Can p be exactly 0 or 1?

Yes, p = 0 means success never happens (all probability mass at k = 0) and p = 1 means success always happens (all probability mass at k = n); the calculator handles both edge cases correctly.

What's the largest n this calculator supports?

Up to 1,000 trials. Beyond that, the exact combinatorial calculation becomes numerically unstable and a normal approximation would be more appropriate.

Where is the binomial distribution used in real life?

It's used in quality control (defective items per batch), clinical trials (patients responding to treatment), A/B testing (conversions per visitor), and any scenario with a fixed number of independent yes/no trials.

Binomial Probability Distribution Calculator — Find Exact and Cumulative Probabilities

Our Binomial Probability Distribution Calculator finds the probability of exactly k successes in n independent trials, each with success probability p, using the binomial probability mass function P(X = k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ. It also reports the cumulative probability P(X ≤ k), the complementary P(X ≥ k), and the distribution's mean (np) and variance (np(1−p)) — the complete toolkit for reasoning about repeated yes/no trials like coin flips, quality-control checks, or A/B test conversions.

Quick Answer

The binomial distribution gives the probability of exactly k successes in n independent trials with success probability p: P(X = k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ. Enter n, p, and k below to get the exact probability, cumulative probability, and mean/variance.

Trials (n)

Probability of success (p)

Successes (k)

Enter n, p, and k, then click Calculate.

How to Use the Binomial Probability Distribution Calculator — P(X=k) Online

1
Enter the number of trials (n).
2
Enter the probability of success on a single trial (p), between 0 and 1.
3
Enter the exact number of successes (k) you want the probability for.
4
Click 'Calculate' to get P(X = k), the cumulative probability P(X ≤ k), and the mean/variance.

Why Use Binomial Probability Distribution Calculator — P(X=k) Online?

The binomial distribution answers 'how likely is exactly k successes out of n independent yes/no trials' — the same question behind coin-flip streaks, defect rates in a batch, and conversion counts in an A/B test. This is distinct from the site's Binomial Coefficient Calculator, which computes C(n,k) alone; here that coefficient is one factor inside a full probability calculation that also weighs the probability of every success and every failure. Multiplying C(n,k) by pᵏ and (1−p)ⁿ⁻ᵏ by hand is tedious and error-prone for anything beyond small n, so this calculator computes the exact probability, its cumulative sum, and summary statistics instantly.

Frequently Asked Questions

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