Randomness and distribution are important concepts underpinning the ability to think and reason probabilistically. Traditional approaches to teaching the Poisson distribution focus on mathematical definitions and formulae which obscure the randomness intrinsic in this process. Advances in technology have made it possible for students learning about probability to model the Poisson process. In this paper we explore the reasoning of six introductory probability students as they interacted with a prototype software designed to visibilize randomness, and to make transparent the link between the Poisson and exponential distributions. We focus on a task involving both real data and simulated data. Our findings highlight the fact that the tool and tasks seem to help students' understanding of the link between the Poisson and exponential distributions, and to gain a deeper appreciation of distribution and randomness.