Variability models are broadly used to specify the configurable features of highly customizable software. In practice, they can be large, defining thousands of features with their dependencies and conflicts. In such cases, visualization techniques and automated analysis support are crucial for understanding the models. This paper contributes to this line of research by presenting a novel, probabilistic foundation for statistical reasoning about variability models. Our approach not only provides a new way to visualize, describe and interpret variability models, but it also supports the improvement of additional state-of-the-art methods for software product lines; for instance, providing exact computations where only approximations were available before, and increasing the sensitivity of existing analysis operations for variability models. We demonstrate the benefits of our approach using real case studies with up to 17,365 features, and written in two different languages (KConfig and feature models).