Building probability models in a differential equations course

An application is considered of elementary differential equations rarely, if ever, used in first courses on the subject. The application is to the derivation of the exponential and Poisson probability distributions. The relationship of continuous and discrete probability models is discussed, and how Poisson model is seen to provide an excellent example of how modelling can move between the two. The development is placed in the context of current trends in modelling in mathematics education.