The computational complexity of winner determination is a classical and important problem in computational social choice. Previous work based on worst-case analysis has established NP-hardness of winner determination for some classic voting rules, such as Kemeny, Dodgson, and Young. In this paper, we revisit the classical problem of winner determination through the lens of semi-random analysis, which is a worst average-case analysis where the preferences are generated from a distribution chosen by the adversary. Under a natural class of semi-random models that are inspired by recommender systems, we prove that winner determination remains hard for Dodgson, Young, and some multi-winner rules such as the Chamberlin-Courant rule and the Monroe rule. Under another natural class of semi-random models that are extensions of the Impartial Culture, we show that winner determination is hard for Kemeny, but is easy for Dodgson. This illustrates an interesting separation between Kemeny and Dodgson.