Abstract: Building on her seminal work regarding moduli space volumes for Riemann surfaces, Mirzakhani also calculated expected values for various geometric functions on moduli space. Notably, she examined the expected Cheeger constant, the injectivity radius at a random point, and the statistical distribution of different types of curves on surfaces of large genus. We will review several of Mirzakhani's key results, which collectively offer insights into the geometry of random surfaces in high genus. Following this, we will explore some extensions of her findings in the context of translation surfaces.