We develop a verification framework aimed at establishing the correctness of discrete sampling algorithms. Within the framework, probabilistic programs are treated as distribution transformers. Inspired by recent work on distributional verification of Markov models, we introduce the notion of (inductive) distributional loop invariants for discrete probabilistic programs. These invariants are embedded in a Hoare-like verification framework that includes proof rules for total and partial correctness. Using our framework, we prove the correctness of two discrete sampling algorithms: the Fast Dice Roller and the Fast Loaded Dice Roller.