Introduction to the Binomial Probability Distribution
In probability theory and statistics, the binomial distribution is a discrete probability distribution that models the number of successes in a sequence of $n$ independent trials, each yielding a binary outcome: success (with probability $p$) or failure (with probability $q = 1 - p$). A classic example is flipping a coin multiple times and counting the number of heads. The binomial distribution is fundamental for quality control, medical trials (determining recovery rates), spam filters, and games of chance.
For a distribution to be binomial, four conditions must be met (often remembered by the acronym BINS):
1. Binary: Trials have only two possible outcomes (success or failure). 2. Independent: The outcome of one trial does not affect the others. 3. Number: There is a fixed, pre-determined number of trials $n$. 4. Same: The probability of success $p$ is constant for each trial.
This calculator evaluates binomial probabilities. By entering the trials $n$, success probability $p$, and target success count $k$, it computes the individual PMF, cumulative CDF, mean, and variance, plotting a complete probability distribution histogram.