Binomial Distribution Calculator - Calculate Probabilities
Calculate binomial probabilities, mean, variance, and cumulative probabilities for any number of trials and success probability
Binomial Parameters
Results
What is a Binomial Distribution Calculator?
A Binomial Distribution Calculator computes probabilities for discrete events with exactly two outcomes (success or failure) over a fixed number of independent trials.
This calculator is used for:
- Quality Control - Defect rates in manufacturing
- Medical Research - Treatment success rates
- Survey Analysis - Yes/no response probabilities
- Genetics - Inheritance probability calculations
To understand continuous probability distributions, explore our Beta Distribution Calculator to model probabilities bounded between 0 and 1 for Bayesian analysis.
To work with another discrete probability distribution, check out our Geometric Distribution Calculator to find probabilities for the number of trials until the first success.
To estimate population parameters from your samples, visit our Confidence Interval Calculator to determine ranges for means and proportions with statistical confidence.
To understand how sample distributions behave, use our Central Limit Theorem Calculator to see how sampling distributions approach normality with larger samples.
How Binomial Distribution Works
The binomial probability formula:
Where:
- n = Number of trials
- k = Number of successes
- p = Probability of success
- C(n,k) = n! / (k! × (n-k)!)
Key Concepts Explained
Independent Trials
Each trial outcome doesn't affect others. Like flipping a coin multiple times.
Fixed Probability
The probability of success remains constant across all trials.
Cumulative Probability
P(X ≤ k) is the sum of probabilities from 0 to k successes.
How to Use This Calculator
Enter Number of Trials (n)
Total number of independent trials or experiments
Enter Number of Successes (k)
Desired number of successful outcomes
Enter Success Probability (p)
Probability of success on each trial (0 to 1)
Benefits of This Calculator
- Complete Analysis - Exact and cumulative probabilities
- Statistical Measures - Mean, variance, standard deviation
- Instant Results - Immediate probability calculations
- Accurate Formulas - Uses standard binomial formulas
- Educational Tool - Perfect for statistics students
- Research Ready - Suitable for academic analysis
Factors Affecting Results
- Sample Size (n) - More trials provide more data points
- Success Rate (p) - Affects distribution shape and spread
- Independence - Trials must not influence each other
- Constant Probability - p must remain same for all trials
- Binary Outcome - Only two outcomes per trial
- Large n Approximation - When np > 5, can use normal approximation
Frequently Asked Questions
What is a binomial distribution?
A binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success.
What is the formula for binomial probability?
The binomial probability formula is: P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and C(n,k) is the combination formula.
What is the mean of a binomial distribution?
The mean of a binomial distribution is: μ = n × p, where n is the number of trials and p is the probability of success.
What is the variance of a binomial distribution?
The variance of a binomial distribution is: σ² = n × p × (1 - p), where n is the number of trials and p is the probability of success.
When do you use binomial distribution?
Use binomial distribution when you have a fixed number of independent trials, each with two possible outcomes (success/failure), and the probability of success remains constant across all trials.
What is the difference between binomial and normal distribution?
Binomial is discrete (counting successes), while normal is continuous. Binomial requires fixed trials with constant probability. As n increases and np > 5, binomial approximates normal distribution.