Binomial PMF Calculator
Compute P(X = k) for a binomial random variable where X ~ Binomial(n, p).
Formula used: P(X = k) = C(n, k) pk (1-p)n-k
What this binomial PMF calculator does
This calculator gives the probability of getting exactly k successes in n independent trials, when each trial has the same success probability p. In statistics, this is called the binomial probability mass function (PMF).
If you're running a repeated yes/no process—like "made sale or no sale," "heads or tails," or "clicked ad or not clicked ad"—this tool is ideal.
How to use the calculator
- n: total number of trials (must be a whole number 0 or greater)
- k: number of successes you want exactly (whole number from 0 to n)
- p: probability of success per trial (between 0 and 1)
After entering your values, click Calculate PMF. The result shows:
- The probability as a decimal
- The equivalent percentage
- The natural log of the probability (useful for very small probabilities)
Example
Scenario: Email response campaign
Suppose you send 10 emails, and each recipient has a 30% chance of responding. What is the probability that exactly 4 people respond?
- n = 10
- k = 4
- p = 0.30
The PMF gives the chance of that exact outcome. Try these values in the calculator above to confirm.
When binomial PMF is appropriate
The binomial model is appropriate when all of the following are true:
- There is a fixed number of trials, n
- Each trial has only two outcomes (success/failure)
- Trials are independent
- The success probability p is constant for every trial
If these assumptions are violated (for example, probability changes from trial to trial), a different model may be better.
Common mistakes to avoid
- Using a non-integer for n or k
- Choosing k greater than n
- Entering p as a percentage (like 30) instead of a decimal (0.30)
- Confusing exact probability P(X = k) with cumulative probability P(X ≤ k)
Quick interpretation tips
What a small PMF value means
A tiny value does not mean the model is wrong. It simply means that the exact count k is unlikely under your assumptions.
Comparing outcomes
To compare likely outcomes, compute PMF values for multiple k values (for example k = 0 through n) and see which are highest.
Final thought
A binomial PMF calculator is a fast way to quantify exact-outcome uncertainty in repeated binary experiments. Whether you're doing A/B testing, quality control, sports analytics, or exam probability problems, this tool gives clean, immediate answers grounded in standard probability theory.