Binomial Distribution Calculator
Find the probability of getting exactly k successes — and up to k, and more than k — in n independent trials, each with success probability p. This calculator also reports the mean, variance, and standard deviation of the binomial distribution.
Formula
P(X=k) = C(n,k) × p^k × (1−p)^(n−k) · μ = n·p · σ = √(n·p·q)
- A binomial distribution models the number of successes in n independent trials when each trial succeeds with the same probability p.
- P(X=k) is the chance of exactly k successes; P(X≤k) sums the probabilities from 0 up to k; P(X>k) is the rest.
- C(n,k) is the binomial coefficient 'n choose k' — the number of ways k successes can occur among n trials.
- The mean (expected number of successes) is μ = n·p, and the variance is σ² = n·p·(1−p).
- p must be between 0 and 1, and k must be between 0 and n; the trials must be independent with a constant success probability.
Example Calculation
Inputs
- Number of Trials (n): 10
- Probability of Success (p): 0.5
- Number of Successes (k): 4
Suppose you enter: Number of Trials (n) = 10, Probability of Success (p) = 0.5, Number of Successes (k) = 4. The calculator applies the formula (P(X=k) = C(n,k) × p^k × (1−p)^(n−k) · μ = n·p · σ = √(n·p·q)) and shows all output values below. Change any input field to immediately see how the result changes.
Frequently asked questions
What is a binomial distribution?
It describes the number of successes in a fixed number of independent trials, where each trial has only two outcomes (success or failure) and the same probability of success.
What is the difference between P(X=k) and P(X≤k)?
P(X=k) is the probability of exactly k successes. P(X≤k) is the cumulative probability of getting k or fewer successes, summing P(X=0) through P(X=k).
When can I use the binomial distribution?
When you have a fixed number of independent trials, each with two outcomes and a constant success probability — such as coin flips, pass/fail tests, or yes/no survey answers.
What do the mean and standard deviation tell me?
The mean μ = n·p is the expected number of successes. The standard deviation σ = √(n·p·q) measures how much the actual count typically varies around that average.
Why must p be between 0 and 1?
p is a probability, so it cannot be less than 0 or more than 1. Values outside that range return an error.