ABSTRACT
This chapter examines the simplest infinite sample spaces which seem to arise naturally. It shows that there is a uniform method which is based on the finite state diagram representation of the problem. This method leads to rational generating functions, from which can easily get both the mean and the variance of the distribution. The chapter introduces the safety device of the conservation of probability; it involves one extra state (which generally do not care about though it may be needed in some problems). It also provides an intuitive way of thinking about problems as if probability were flowing water. The conservation of probability allows us to think of the probability as an incompressible fluid, like water. Probability is no exception; probability arose long after the formation of the classical mathematics and it is not obvious that we should try to use the unmodified mathematics in all applications.
