Negative Binomial Distribution Calculator

The negative binomial distribution describes how many failures occur before a fixed number of successes is reached in a sequence of independent trials. Every trial has only two outcomes, conventionally called success and failure, and the probability of success remains constant. This calculator uses r for the required number of successes, p for the success probability on each trial, and k for the number of failures observed before the rth success. That definition is common in probability courses, quality control, reliability studies, epidemiology, and count-data modeling. The exact probability is found with the probability mass function C(k + r - 1, k) times p raised to r times (1 - p) raised to k. The combination counts the possible arrangements of earlier successes and failures while requiring the final trial to be the target success. The calculator also sums the mass function from zero through k to report P(X <= k), then subtracts that cumulative value from one to report P(X > k). Together these values answer exact, at-most, and more-than questions without separate calculations. The expected number of failures is r(1 - p) / p, and the variance is r(1 - p) / p squared. A larger success probability generally moves the distribution toward zero failures and reduces its spread. A larger target number of successes increases both the expected failures and the variance. When p equals one, every trial succeeds, so zero failures has probability one and the mean and variance are both zero. Use whole numbers for r and k because they count outcomes. Enter p as a decimal between zero and one, such as 0.35 for a 35 percent chance. Trials must be independent and identically distributed for the standard model to apply. If probabilities change between trials, outcomes influence one another, or the process stops for another reason, a different model may be more appropriate. Also note that some textbooks define the negative binomial variable as the total number of trials rather than failures. Under that convention, total trials equals k + r, while the probability itself is unchanged after the variables are translated correctly.