This 30-minute talk will describe a serendipitous appearance of notions from statistics  (e.g., quantiles, tail distributions, ccdfs) in the study of fundamental thresholds in algebraic geometry. Two unexpected twists in the plot are provided by a discrete geometry version of Riemannian collapsing blow-up analysis, and by the (miraculous) appearance of Weierstrass gap theory. Based on joint work with C. Jin and G. Tian.