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A discrete distribution giving the probability of k successes in n independent Bernoulli trials with constant success probability p.
DESCRIPTION
The binomial distribution models counts of successes over a fixed number of independent trials with identical success probability. Its parameters are n (trials) and p (success probability). The mean is np and variance is np(1−p). It underlies confidence intervals for proportions and A/B testing. For large n, a normal approximation may apply. Conjugacy with the beta distribution simplifies Bayesian inference.
ALTERNATIVE DEFINITIONS
Wikipedia
The binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome.
Classical definition emphasizing independent Bernoulli trials and success counts.
NIST/SEMATECH e-Handbook
The binomial distribution is the discrete probability distribution for the number of successes in a sequence of n Bernoulli trials, where the probability of success is the same for each trial.
Adds the assumption of identical success probability across trials.
Khan Academy
The binomial distribution gives the probability of getting k successes in n independent Bernoulli trials with probability p of success on each trial.
Educational, succinct formulation with k, n, and p.
RELATED TERMS
Bernoulli Distribution Probability Mass Function Normal Approximation Hypothesis Testing Confidence Interval Beta Distribution p-value Central Limit Theorem
COMPETING TERMS
RELATIONS
- Derived from: Bernoulli Distribution
- Narrower: Bernoulli Distribution Negative Binomial Distribution
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist Data Analyst AI Engineer Information Analyst
EXTERNAL SOURCES
- Wikipedia - Binomial distribution
- Wolfram MathWorld - Mathematical properties and formulas
- Statlect - Detailed tutorial and derivations
- Penn State STAT 414 - Course notes on binomial
TOP QUESTIONS
When is the Binomial Distribution appropriate?
Use it for a fixed number of independent trials with identical success probability and two outcomes per trial.
Category: Understanding
How is the Binomial Distribution used in A/B testing?
It models conversion counts; confidence intervals and hypothesis tests on proportions often assume binomial sampling.
Category: Example
What are common mistakes with the Binomial Distribution?
Assuming independence or constant probability when they don’t hold, leading to underestimated variance and overconfident inferences.
Category: Mistakes
How to approximate the Binomial Distribution with a normal?
Use N(np, np(1−p)) when n is large and p not too close to 0 or 1; apply continuity correction for better accuracy.
Category: Methods
What are the mean and variance of the Binomial Distribution?
Mean is np and variance is np(1−p). These summarize expected successes and variability.
Category: Understanding
How does the Beta distribution relate to Binomial?
Beta is conjugate prior for p, yielding Beta-Binomial posteriors in Bayesian analyses.
Category: Understanding
What’s a real-world example of Binomial modeling?
Counting defective items in a lot when each item independently fails with probability p.
Category: Example
How do overdispersion issues arise with Binomial?
If trials are correlated or p varies, variance exceeds np(1−p); models like Beta-Binomial address this.
Category: Risks
How to estimate p from Binomial data?
The maximum likelihood estimator is k/n; interval estimation can use Wilson or Clopper–Pearson intervals.
Category: Methods