Binomial PMF calculator

The probability mass function, or PMF, answers one specific question. Out of n trials, what is the probability of exactly k successes? It excludes counts below and above k.

For example, exactly 2 successes includes 2 only. If your question says “2 or fewer,” you need the cumulative distribution function, which adds several possible counts.

Some references call this a binomial PDF. Because a binomial count is discrete, PMF is the more precise term. This page calculates probability mass; it does not create PDF files or give device-specific calculator commands.

P(X = k) = C(n,k) p^k (1−p)^n−k

n is the number of trials. p is the probability of success on each trial. k is the number of successes. Use a fixed n, independent trials, two outcomes per trial and one shared success probability.

The factor C(n,k) = n! / [k!(n−k)!] counts how many arrangements contain k successes. The factor pk(1−p)n−k is the probability of one such arrangement. Multiplying counts all of them.

After you calculate, open “Show the calculation steps” to see the combination count and the formula with your actual values. The core calculation avoids forming large factorials, so n = 1000 does not overflow a factorial.

This is a separate mathematical example, not a prefilled result. In four independent trials with success probability 0.5, find exactly two successes.

1. Count the arrangements. C(4,2) = 4! / (2! × 2!) = 6.
2. Find the probability of each arrangement. 0.5² × 0.5² = 1/16.
3. Multiply. P(X = 2) = 6/16 = 0.375 = 37.5%.

A complete check uses the PMF weights 1, 4, 6, 4, 1 for k = 0 through 4, each divided by 16. They add to 1. The calculator’s table and PMF graph show the same distribution for these inputs.

The mean np is the expected success count, which need not be an integer. Variance np(1−p) describes spread in squared count units. Standard deviation is its square root, in success-count units. They describe the whole distribution, not the probability of the selected event.

On the PMF graph, bar height is the probability of one count. A tiny tail may be invisible on a linear axis while still appearing as a nonzero scientific-notation value in the table.

At n = 0, exactly zero successes has probability 1. At p = 0, only zero successes can occur. At p = 1, only n successes can occur. The tool rejects fractional counts and k outside 0 through n.

Keep p as a decimal. A 20% success chance is 0.2, not 20. “Success” simply names the outcome you are counting; it does not have to mean a desirable result.

If the question asks for “at least” or “more than,” check which boundary is included. If you are unsure how to choose n, p and k, use the step-by-step calculation guide.

Formula reference NIST’s binomial distribution handbook.