The geometric distribution describes the probability of observing the first success on the nth
trial. The negative binomial distribution is more general: it describes the probability of observ- ing the kth success on the nth trial.
EXAMPLE
Each day a high school football coach tells his star kicker, Brian, that he can go home after he successfully kicks four 35 yard field goals. Suppose we say each kick has a probability p of being successful. If p is small – e.g. close to 0.1 – would we expect Brian to need many attempts before he successfully kicks his fourth field goal?
We are waiting for the fourth success (k = 4). If the probability of a success (p) is small, then the number of attempts (n) will probably be large. This means that Brian is more likely to need many attempts before he gets k = 4 successes. To put this another way, the probability of n being small is low.