the negative binomial (Pascal) distribution

The negative binomial distribution generalizes the geometric from 'wait for the first success' to 'wait for the r-th success'. You keep running independent Bernoulli(p) trials and stop only when successes number r. The total number of trials this takes is random and follows the negative binomial law. Setting r = 1 recovers the geometric exactly — the geometric is the negative binomial's special case.

Take the 'number of trials' convention. To have the r-th success land on trial k, two things must hold simultaneously: the very last trial is a success, and among the first k - 1 trials there were exactly r - 1 successes. That gives the pmf P(X = k) = C(k - 1, r - 1) p^r (1 - p)^(k - r) for k = r, r + 1, ... The C(k - 1, r - 1) counts the orderings of the earlier successes (the last slot is fixed as a success, so we do not let it float). Because the process is just r independent geometric waits stacked end to end, the mean is simply r times the geometric mean, E[X] = r/p, and the variance is r(1 - p)/p^2. When integer r is generalized to any positive real, the combinatorial coefficient becomes a gamma-function expression, and the distribution doubles as a flexible, overdispersed alternative to the Poisson for count data whose variance exceeds its mean.

Watch the conventions again: some texts count failures rather than trials, and the gamma-generalized version (sometimes called the Polya distribution) is parameterized differently still. The name 'negative binomial' comes from the fact that its probabilities are terms of a binomial series with a negative exponent — a bookkeeping curiosity, not a sign that anything is negative.