The binomial distribution calculator computes exact probabilities for discrete experiments with a fixed number of independent Bernoulli trials. Given n trials and probability p, find P(X=k), cumulative P(X≤k), P(X≥k), mean, variance, and standard deviation.
Binomial Distribution
Step-by-Step for P(X=k)
Formula Reference
How to Use the Binomial Distribution Calculator
The binomial distribution applies to experiments with a fixed number of independent trials, each with two outcomes (success/failure) and constant probability p of success.
Step 1: Enter Parameters
Enter n (number of trials), p (probability of success on each trial, 0 to 1), and k (number of successes you want to compute the probability for).
Example: Quality Control
A factory produces parts where 5% are defective (p=0.05). In a sample of n=20 parts, what's the probability of exactly k=2 defects? P(X=2) = C(20,2) × 0.05² × 0.95¹⁸ = 190 × 0.0025 × 0.3972 = 0.1887, or about 18.9%.
Cumulative Probabilities
P(X≤k) is the probability of at most k successes. P(X≥k) is the probability of at least k successes. These are computed by summing P(X=i) for all valid i. Useful for questions like "probability of passing" (at least 60% correct).
When to Use Binomial
Fixed n, independent trials, binary outcome, constant p. Common examples: coin flips, multiple choice guessing, product defects, free throws in basketball, survey responses (yes/no).
FAQ
What is the binomial distribution?
The binomial distribution models the number of successes in n independent trials, each with probability p of success. It applies when: you have a fixed number of trials, each trial is independent, each trial has only two outcomes, and the probability p is constant.
What is the binomial probability formula?
P(X=k) = C(n,k) × p^k × (1-p)^(n-k), where C(n,k) = n! / (k!(n-k)!) is the binomial coefficient. This counts the number of ways to choose k successes from n trials, times the probability of each specific arrangement.
What are the mean and variance of a binomial distribution?
The mean (expected value) is μ = np. The variance is σ² = np(1-p). The standard deviation is σ = √(np(1-p)). Example: flipping a fair coin 100 times has mean = 50 heads, variance = 25, σ = 5.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
When should I use binomial vs normal distribution?
Use binomial for exact probability calculations with small n. For large n (generally n > 30) where np > 5 and n(1-p) > 5, the normal distribution with μ=np and σ=√(np(1-p)) gives a good approximation (Central Limit Theorem).