Negative Binomial Distribution Calculator

The negative binomial distribution models the number of trials needed to achieve a specified number of successes. Let k represent the target number of successes, n represent the total number of events (trials), and p represent the probability of success on any given trial. The formula computes the likelihood of observing the k-th success exactly on the n-th trial using the equation: P(X = n) = C(n - 1, k - 1) * p^k * (1 - p)^(n - k), where C(n - 1, k - 1) represents the binomial coefficient calculated as combinations of n - 1 items taken k - 1 at a time.

Imagine you run a specialized sales team where the historical probability of closing a deal on any given pitch is 20%, or p = 0.20. You want to calculate the exact probability that your 3rd successful deal (k = 3) occurs precisely on your 10th pitch (n = 10). First, find the combinations of prior trials: C(10 - 1, 3 - 1) = C(9, 2) = 36. Next, apply the probability of successes: 0.20^3 = 0.008. Then, calculate the remaining failures: (1 - 0.20)^(10 - 3) = 0.80^7 = 0.2097152. Multiplying these components together yields 36 * 0.008 * 0.2097152 = 0.060416, meaning there is roughly a 6.04% chance that your third closed deal happens on the tenth pitch.

To ensure accurate statistical modeling, always verify that your trials are independent and that the probability of success remains constant across every event. Double-check that your total number of events (n) is greater than or equal to your target number of successes (k), as it is mathematically impossible to achieve multiple successes in fewer trials. Additionally, ensure your success probability is expressed as a decimal between 0 and 1 rather than a percentage.