Negative Binomial Distribution Calculator
Calculate the probability of needing exactly k trials to achieve r successes using the negative binomial distribution. This distribution models the number of trials required before reaching a fixed number of successes.
The negative binomial distribution generalizes the geometric distribution. While the geometric distribution counts trials until the first success, the negative binomial counts trials until the r-th success.
Formula: P(X = k) = C(k-1, r-1) x p^r x (1-p)^(k-r), where k is the total number of trials, r is the desired number of successes, and p is the probability of success per trial. The combination C(k-1, r-1) counts the ways to arrange r-1 successes in the first k-1 trials (the k-th trial must be a success).
Key properties: E(X) = r/p is the expected number of trials. Var(X) = r(1-p)/p^2. As r increases, the distribution becomes more symmetric and approaches a normal shape. Applications include modeling the number of sales calls needed to close r deals, the number of patients to screen before finding r eligible participants, or the number of wells to drill before finding r productive ones.