Binomial Distribution (PMF) Calculator
Probability of exactly k successes in n trials.
Formula first
Overview
The Binomial Distribution Probability Mass Function calculates the likelihood of achieving exactly k successes in a fixed number of independent Bernoulli trials. It assumes two possible outcomes for each trial and a constant probability of success throughout the entire process.
Symbols
Variables
n = Trials, k = Successes, p = Success Probability, P = Probability
Apply it well
When To Use
When to use: Use this equation when the experiment consists of n identical trials, each with only two possible outcomes such as success or failure. It requires that the trials are independent and that the probability of success remains unchanged from one trial to the next.
Why it matters: This distribution is a cornerstone of statistical inference, used extensively in quality control to estimate defect rates and in clinical trials to determine drug efficacy. It provides a mathematical framework for predicting outcomes in any binary system where randomness is present.
Avoid these traps
Common Mistakes
- Using permutations instead of combinations.
- Forgetting (1-p)^(n-k).
- Not restricting k to 0..n.
One free problem
Practice Problem
A professional basketball player has a 70% success rate for free throws. If they take 5 shots, what is the probability that they make exactly 3 of them?
Solve for: P
Hint: Calculate the number of ways to choose 3 shots out of 5, then multiply by the probability of 3 successes and 2 failures.
The full worked solution stays in the interactive walkthrough.
References
Sources
- A First Course in Probability by Sheldon Ross
- Wikipedia: Binomial distribution
- Probability and Statistics for Engineers and Scientists by Walpole, Myers, Myers, Ye
- IUPAC Gold Book: Binomial distribution