Binomial Distribution Calculator
Calculate probabilities for binomial experiments with our easy-to-use calculator. The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with success probability p.
Binomial Distribution Parameters
- Mean (μ): μ = n × p
- Variance (σ²): σ² = n × p × (1 - p)
- Standard Deviation (σ): σ = √(n × p × (1 - p))
Probability: 0.2461
Distribution Parameters
n = 10, p = 0.5, k = 5
Probability Details
| Description | Value |
|---|---|
| Probability Mass Function (PMF) | 0.2461 |
| Cumulative Probability (CDF) | 0.6230 |
| Complementary CDF | 0.3769 |
About the Binomial Distribution
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials (experiments with exactly two possible outcomes: success or failure) with the same probability of success.
Key Characteristics
- Fixed number of trials (n): The experiment consists of n identical trials.
- Independent trials: The outcome of any trial doesn't affect other trials.
- Two possible outcomes: Each trial results in success (with probability p) or failure (with probability 1-p).
- Constant probability: The probability of success (p) remains the same for each trial.
Probability Mass Function
The probability of getting exactly k successes in n trials is given by:
P(X = k) = C(n,k) × pk × (1-p)n-k
Where C(n,k) is the binomial coefficient "n choose k" calculated as:
C(n,k) = n! / (k! × (n-k)!)
When to Use the Binomial Distribution
The binomial distribution is appropriate when:
- Counting occurrences of an event in a fixed number of trials
- Each trial has only two possible outcomes (success/failure)
- Trials are independent of each other
- Probability of success remains constant across trials
Examples
- Number of heads in 10 coin flips
- Number of defective items in a batch of 100
- Number of patients responding to a treatment in a clinical trial