The Binomial Setting

Whenever a process consists of $n$ independent trials with the same success probability $p$, we can model the number of successes with a binomial random variable. Each trial has two outcomes, often termed "success" and "failure." Examples include the number of heads when flipping a coin or the number of defective items in a batch. Understanding the probability distribution of this count is fundamental in statistics and decision making.

Probability Mass Function

The probability of observing exactly $k$ successes is given by the formula

$P\left(k\right)=\frac{\text{n choose k}}{1}{p}^k{1-p}^{n-k}$

where the binomial coefficient counts the number of distinct ways to pick $k$ successes out of $n$ trials.

Cumulative Probability

The cumulative distribution function (CDF) returns the probability that the number of successes is less than or equal to $k$. Evaluating the CDF involves summing the probability mass over all values from zero up to $k$. This gives a measure of how likely we are to observe at most $k$ successes in $n$ attempts.

Usage Notes

Input the number of trials, the probability of success for each trial, and the desired count $k$. The calculator outputs the probability mass function value and the cumulative probability. These results are handy for quality control, reliability analysis, and any setting in which repeated binary outcomes arise.