binomial pdf calculator

Binomial PDF (PMF) Calculator

Use this calculator to find the exact probability of getting exactly k successes in n independent trials, where each trial has success probability p.

Formula: P(X = k) = C(n, k) p^k (1 - p)^{n - k}

What this binomial PDF calculator does

This binomial distribution calculator computes the probability of observing an exact number of successes in a fixed number of trials. In strict statistics language, this is a PMF (probability mass function), but many people search for “binomial PDF calculator,” so that is the phrase used here.

If you are working on exam prep, A/B testing, quality control, reliability, clinical trials, or basic probability homework, this tool gives a quick, accurate result without needing a separate statistics package.

When the binomial model is appropriate

The binomial model fits situations where all of the following are true:

You have a fixed number of trials, n.
Each trial has only two outcomes (success or failure).
The probability of success, p, stays the same each trial.
The trials are independent.

If one or more of these assumptions fail, you may need a different model (for example, hypergeometric, Poisson, or negative binomial).

How to use this calculator

Enter n: total number of trials.
Enter p: probability of success on each trial (between 0 and 1).
Enter k: exact number of successes you want to evaluate.
Click Calculate to see the exact probability P(X = k).

The result area also shows cumulative values P(X ≤ k) and P(X ≥ k), plus the mean and standard deviation for the same binomial setup.

Interpreting the output

Exact probability: P(X = k)

This is the chance of getting exactly k successes, not “at least” or “at most.” It is often used in scenario-specific questions such as: “What is the probability exactly 6 out of 20 customers click?”

Cumulative probability: P(X ≤ k)

This adds up all probabilities from 0 through k. It is useful when the question asks for “at most k successes.”

Upper-tail probability: P(X ≥ k)

This gives the probability of getting k or more successes. It is useful for “at least k” style questions.

Worked binomial example

Suppose a manufacturing process has a defect probability of 0.08 per part. You inspect 30 parts and want the probability of exactly 2 defects. Set n = 30, p = 0.08, and k = 2. The calculator returns P(X = 2), which gives the exact probability of that event.

You can then compare this with P(X ≤ 2) to ask, “How likely is it that I see at most 2 defects?” This is often more useful for process-control decisions.

Common mistakes to avoid

Entering p as a percent (e.g., 30) instead of a decimal (0.30).
Using a non-integer value for n or k.
Entering k > n, which is impossible and has probability 0.
Confusing “exactly k” with “at most k” or “at least k.”

FAQ

Is this really a PDF?

For discrete distributions like binomial, the formal term is PMF. Many learners still use “PDF calculator” in search queries, so both terms are commonly seen.

Can I use large values of n?

Yes, this calculator uses stable logarithmic math for the exact probability. Extremely large values may still be limited by floating-point precision in any browser.

What are the mean and variance of a binomial variable?

Mean: μ = np. Variance: σ² = np(1 - p). Standard deviation is the square root of that variance.