Who has the best probabilities? Luck versus skill in prediction tournaments

Abstract: An informal and elementary introduction to probability scoring and forecast verification and improvement, slightly extended from Significance 22:3(2025)16, which might be useful for less mathematical readers as a prologue to the classic review by Gneiting and Raftery [Strictly proper scoring rules, prediction, and estimation, Journal of the American Statistical Association 102 (2007): 359].