The Poisson distribution describes the probability of a certain number of events occurring in an interval of time when the occurrence of the individual events is independent of one another and the events occur with a fixed mean rate. Probably the best-known example of the Poisson distribution in the physics curriculum is the temporal distribution of counts in nuclear or photon counting experiments. Although the Poisson distribution is derived in various physics books, these derivations are at an advanced level and rely on a background in statistics. The aim of this note is to start at the most elementary idea of probability and thread a path through the mathematics needed for the derivation of the probability density. Although none of what is described here is novel, it is entirely self-contained and should be a useful resource for anyone teaching or utilizing the Poisson distribution. A computer simulation is also described that uses the uniform random deviates available in high-level computer languages to generate sequences of events, which are then analyzed and shown to have a Poisson distribution. The material presented here has been used as a handout for students taking University Modern Physics lab. This supplements their studies of error analysis from the book by Taylor.