Precipitation processes, and their simulation in weather and climate models, are notoriously complex. We rely on climate models for projections of how statistics of precipitation will change in a warming climate. What do we do when models exhibit deficiencies in simulation of probability distributions of precipitation in current climate and their relationship to the water vapor and temperature environment? Here we argue that many of these precipitation statistics can be captured by conceptually simple models, based on economical assumptions, providing an understanding of the processes that create the characteristic shapes of probability distributions for different precipitation measures. These include event accumulations, daily-average intensities and the size of spatial clusters. An overview will be provided of the dialogue between insights from the simple models and diagnostics of observations and complex model simulations.