Binomial Distribution Calculator
Binomial Distribution Calculator
Calculate probabilities for binomial distributions with n trials and probability p
Binomial Distribution Calculator: Complete Statistical Guide
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials.Each trial has the same probability of success, making it ideal for modeling scenarios like coin flips, quality control tests, survey responses, and clinical trials.
Our professional binomial distribution calculator provides probability analysis including exact probabilities, cumulative distributions, range calculations, and complete statistical measures like mean, variance, and distribution characteristics.
Quick Answer
For binomial distribution: You need number of trials (n), probability of success (p), and number of successes (k). The probability of exactly k successes is P(X = k) = C(n,k) × p^k × (1-p)^(n-k). For example, the probability of getting exactly 3 heads in 5 coin flips is approximately 31.25%.
Mathematical Foundation
Binomial probability formula where C(n,k) is combinations, p is success probability, and (1-p) is failure probability
Key Distribution Parameters:
Mean (Expected Value)
μ = n × p. The expected number of successes in n trials with probability p.
Variance
σ² = n × p × (1-p). Measures the spread of the distribution around the mean.
Standard Deviation
σ = √(n × p × (1-p)). Square root of variance, measuring typical deviation from the mean.
Common Applications
Quality Control & Manufacturing
Defect Analysis
Model number of defective items in production batches
Pass/Fail Testing
Analyze success rates in component testing
Medical & Clinical Research
Drug Efficacy
Model patient response rates to treatments
Clinical Trials
Analyze success rates in medical interventions
Example Problems with Solutions
Example 1: Coin Flipping
What's the probability of getting exactly 3 heads in 5 fair coin flips?
Answer: 31.25% chance of exactly 3 heads
Example 2: Quality Control
In a batch of 100 items with 5% defect rate, what's the probability of finding 3 or fewer defects?
Answer: 25.78% chance of 3 or fewer defects
Frequently Asked Questions
When should I use the binomial distribution?
Use binomial distribution when you have: (1) Fixed number of trials, (2) Each trial has only two outcomes (success/failure), (3) Probability of success is constant, and (4) Trials are independent. Common examples include coin flips, quality control, and survey responses.
What's the difference between exact and cumulative probability?
Exact probability P(X = k) is the chance of getting exactly k successes.Cumulative probability P(X ≤ k) is the chance of getting k or fewer successes. Use exact for specific outcomes, cumulative for "at most" or "at least" questions.
How do I interpret the mean and standard deviation?
The mean (μ = np) tells you the expected number of successes. The standard deviationindicates variability - smaller values mean results cluster near the mean, larger values indicate more spread. About 68% of outcomes fall within one standard deviation of the mean.
Related Statistical Tools
Binomial Distribution Calculator
Calculate probabilities for binomial distributions with n trials and probability p