We live in a world of variability, uncertainty and unpredictability. These factors influence the manner in which computer scientists evaluate the performance of complex systems, biophysicists analyze the behavior of molecules transitioning across cell membranes, and economists treat the ups and downs of financial markets. In these cases and a multitude of others, stochastic processes are used to model the effects of unpredictable forces that operate over intervals of time. Although stochastic models have proven highly successful in many fields, they also present a fundamental dilemma. The more deeply one delves into the mathematical assumptions upon which these models are based, the less likely it seems that real world systems will actually satisfy the required assumptions. How then can one explain the remarkable success of stochastic models? This talk uses a simple and easily understood example to demonstrate that many classical results from stochastic modeling are valid under assumptions that are significantly weaker than generally believed. At the same time, the example also demonstrates that other results, especially those dealing with the occurrence of rare events, are beyond the scope of these weaker assumptions. These “trans-distributional” results are less likely to be predicted accurately by probabilistic models. Some implications of this work for understanding the relationship between variability and randomness will be examined. In addition, applications to the analysis of computer system performance and the calibration of Monte Carlo simulations will be presented in the final sections of this talk.