Binomial Probability Calculator
Find the probability of exactly k successes in n independent trials.
Binomial probability formula
P(X=k) = C(n,k) × p^k × (1−p)^(n−k). It applies to independent trials with the same probability of success.
How Binomial Probability Works
The binomial distribution models the number of successes in a fixed number of independent trials when each trial has the same probability of success. Typical examples include pass/fail outcomes, yes/no responses and defect/no-defect inspections.
Binomial Probability Example
If a fair coin is flipped 10 times and success is defined as heads, the probability of exactly 5 heads can be calculated using n = 10, k = 5 and p = 0.5.
When the Binomial Model Fits
Use this model when the number of trials is fixed, outcomes can be classified into two categories, trials are independent and the success probability remains constant from trial to trial.
Frequently Asked Questions
What does n represent?
It is the total number of trials.
What does k represent?
It is the number of successes of interest.
Can p be greater than 1?
No. Probability must be between 0 and 1.
Binomial Model Assumptions
The binomial model assumes a fixed number of trials, two outcomes per trial, a constant success probability and independent trials. If success probability changes between trials or observations are dependent, a binomial probability may not represent the process accurately.
Methodology, Assumptions and Interpretation
Binomial probability assumes a fixed number of independent trials, two outcomes per trial and a constant success probability. When probability changes between trials or outcomes are dependent, the binomial model may not be appropriate.