Scientists and engineers often use Poisson's probability distribution to characterize the statistics of rare events whose average number is small. Using it correctly is crucial if we are to validate claims of discovery of new phenomena, such as a new fundamental particle (few candidate collision events among millions), a remote galaxy (few photons in the telescope among the billions emitted), or brain damage from using cell phones (few tumors among millions of users). In risk assessment, such as estimating the chance of dying from a horse kick if you're in the Prussian army or from suicide (two of its early uses), it plays a crucial role, which should interest actuaries as well as morticians. The author has noticed that the Poisson distribution is often misunderstood and misapplied, so he describes some of its interesting and relevant properties. He emphasizes visualizing what's happening in the mathematics by using Mathematica.