The purpose of this paper is to discuss the probability distribution that arises when the probability of success at any trial depends linearly upon the number of previous successes. Such a scheme has obvious uses in both biological and economic fields. It will be shown that by assuming a simple linear relationship between the number of previous successes and the probability of success in the next trial, we can derive a distribution that is reasonably easy to handle, provides as good a fit as more usual distributions, and has parameters which are capable of easy physical interpretation. Moreover, for appropriate values of the parameters the negative binomial and the Gram-Charlier systems can be shown to be close approximations.