Consider a sequence of independent trials with the same probability p of success in each trial. The number of trials up to and including the rth success has the negative Binomial distribution with parameters n and r. If the random variable N has the negative binomial distribution with parameters n and r, then

P(N=k) = k₋₁Cr−1 × pr × (1−p)k−r,

for k = r, r+1, r+2, …, and zero for k < r, because there must be at least r trials to have r successes. The negative binomial distribution is derived as follows: for the rth success to occur on the kth trial, there must have been r−1 successes in the first k−1 trials, and the kth trial must result in success. The chance of the former is the chance of r−1 successes in k−1 independent trials with the same probability of success in each trial, which, according to the Binomial distribution with parameters n=k−1 and p, has probability

k₋₁Cr−1 × pr−1 × (1−p)k−r.

The chance of the latter event is p, by assumption. Because the trials are independent, we can find the chance that both events occur by multiplying their chances together, which gives the expression for P(N=k) above.