Calculate binomial distribution probabilities: P(X=k), P(X<=k), P(X>=k), and cumulative values. Get mean, variance, and distribution charts.
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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.
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
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Essential for statistical hypothesis testing
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.
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.
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.
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.