Tom Hutchcroft and I have been working to develop a general theory of percolation on arbitrary finite transitive graphs. We have had to apply tools from the study of percolation on infinite transitive graphs (traditionally studied by "probabilists") to percolation on high-degree finite graphs (traditionally studied by "combinatorialists") and vice versa. In this presentation, I would like to advertise a tool that we have brought back from the high-degree setting. It is a new way to apply a sharp threshold result originally due to Bourgain then sharpened by Hatami. I will explain how this tool can be applied to give a very short proof of a new result in the setting of percolation on infinite transitive graphs, namely that in the supercritical phase, the infinite cluster density `theta' is determined by the (Benjamini-Schramm) local geometry of the underlying graph.