We quantify Prokhorov's Theorem by establishing an explicit formula for the Hausdorff measure of non-compactness (HMNC) for the parametrized Prokhorov metric on the set of Borel probability measures on a Polish space. Furthermore, we quantify the Arzelà-Ascoli Theorem by obtaining upper and lower estimates for the HMNC for the uniform norm on the space of continuous maps of a compact interval into Euclidean N-space, using Jung's Theorem on the Chebyshev radius. Finally, we combine the obtained results to quantify the stochastic Arzelà-Ascoli Theorem by providing upper and lower estimates for the HMNC for the parametrized Prokhorov metric on the set of multivariate continuous stochastic processes.