How long is the wait for an uncommon event?

If the probability of an event is known, the lengths of series until at least 1, 2, ...,r events have taken place in this series are predictable. They have a negative binomial distribution. The use of this distribution is described to predict the lengths of series of patients to be treated with a certain drug until at least 1, 2,...,5 times a patient with for example, the side-effect exanthema will be seen, if an exanthema occurs in 1 out of 100 of these patients. In other examples the events are deaths after an anesthetic, cervical carcinomas in the screening procedure, and significances in statistical tests with alpha = 0.05 as a level of significance.