A growing literature generalizes the classical model of random choice from Theorem 3 of Luce’s monograph (1959) to accommodate zero probabilities. I establish the close connection between this literature and the very next result (Theorem 4) from Luce’s monograph. In the process, I answer some long-standing questions about Luce’s approach to zero probabilities; and generalize almost all of the results from the recent literature on extended Luce rules.