What's the difference between P(X=k) and P(X≤k)?

P(X=k) is the probability of exactly k successes. P(X≤k) is the cumulative probability of k or fewer successes (0, 1, 2, ..., k). Use P(X≤k) when you want 'at most k' successes.

How do I calculate P(X≥k)?

P(X≥k) equals 1 - P(X≤k-1). This gives the probability of k or more successes. Use this for 'at least k' calculations.

What do mean and variance tell me about the distribution?

The mean (μ = np) tells you the expected number of successes. The variance (σ² = np(1-p)) measures the spread. Higher variance means outcomes are more spread out from the mean.

Statistics

Binomial Distribution Calculator

Calculate binomial distribution probabilities, including P(X=k), P(X≤k), P(X≥k), and cumulative probabilities. Get mean, variance, and full probability distribution charts.

Number of Trials (n)

Probability of Success (p)

Calculation Type

Number of Successes (k)

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Binomial Distribution Calculator - Probability Calculator

Calculate binomial distribution probabilities instantly. Find exact, cumulative, and range probabilities for any number of trials and success probability. Includes distribution charts, probability tables, and step-by-step calculations.

What is the Binomial Distribution?

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It's used when there are exactly two outcomes (success/failure), trials are independent, and probability stays constant. Examples include coin flips, quality control inspections, and medical trials.

Binomial Probability Formula

P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

How to Use This Calculator

Binomial Distribution Applications

Quality Control

Calculate probability of defects in manufacturing batches to set acceptance criteria.

Medical Trials

Analyze treatment success rates and determine statistical significance.

Marketing Analytics

Model conversion rates and predict campaign performance.

Sports Statistics

Calculate winning probabilities and game outcome distributions.

Frequently Asked Questions

The binomial distribution requires: (1) Fixed number of trials n, (2) Each trial has only two outcomes (success/failure), (3) Trials are independent, and (4) Probability of success p is constant for each trial.