The comparison of partition and random sampling methods for software testing has received considerable attention in the literature. A standard criterion for comparisons between random and partition testing based on their expected efficacy in program debugging is the probability of detecting at least one failure causing input in the program's domain. We investigate the relative effectiveness of partition testing versus random testing through the powerful mathematical technique of majorization, which was introduced by Hardy et al. (1952). The tools of majorization and the concepts of Schur (convex and concave) functions (1923) enable us to derive general conditions under which partition testing is superior to random testing and, consequently, to give further insights into the value of partition testing strategies.