Erik Curiel explains the challenges in making assessments of probability by making assumptions about the space of universes--or cosmological models--that our theories allow. Erik Curiel (Munich Center for Mathematical Philosophy) explains the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects (e.g., spacetimes) living in infinite-dimensional spaces, working through several examples from cosmology. He concludes that most standard forms of argument used in cosmology to estimate the likelihood of the occurrence of various properties or behaviors of spacetimes have serious mathematical, physical and conceptual problems. This lecture was conducted at the Lindeman Lecture Theatre, Clarendon Laboratory, Oxford, on October 6, 2016.