Negative Binomial Distribution Calculator

The negative binomial counts failures before achieving a set number of successes.

How the Math Works

The Negative Binomial Distribution Calculator uses a fundamental probability formula to determine the likelihood of experiencing a specific number of failures before achieving a set number of successes. The calculation P = C(k+r-1, k)·pʳ·(1-p)ᵏ combines three key components: the binomial coefficient C(k+r-1, k) which counts the number of ways to arrange k failures among k+r total trials, the probability of success p raised to the power of r successes, and the probability of failure (1-p) raised to the power of k failures. This mathematical framework captures all possible sequences where the final trial must be a success, making it distinct from other distribution models.

Practical Applications

This calculator is essential for quality control analysis, where you might want to know the probability of observing k defective items before finding r acceptable products in a manufacturing process. In clinical research, it helps determine the likelihood of treating k patients before achieving r successful outcomes, such as symptom relief. Market researchers use it to calculate the probability of encountering k dissatisfied customers before finding r satisfied ones in a survey. The formula also applies to sports analytics, where you might calculate the probability of a basketball player missing k shots before making r successful shots.

Day-to-Day Use

Understanding this distribution helps you make informed decisions about risk and probability in everyday situations. When shopping, you might use it to estimate how many stores you'd need to visit (failures) before finding the one with the best deal (successes). In personal finance, it can help model the probability of experiencing k financial setbacks before achieving r positive investment outcomes. Sports fans can apply it to understand their team's performance patterns, such as the likelihood of losing k games before winning r crucial playoff games. It's also useful for planning purposes, like estimating how many attempts you might need before completing a DIY project successfully.

Worked example

r 3, p 0.5, k 2 → P ≈ 0.1875.

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