In this paper, we introduce a new one-step iteration process in Banach space and prove the existence of a common random fixed point of three non-expansive multivalued random operators through strong and weak convergences of an iterative process. The necessary and sufficient condition for the convergence of a sequence of measurable functions to a random fixed point of non-expansive multivalued random operators in uniformly convex Banach spaces is also established. Our random iteration scheme includes new random multivalued iterations as special cases. The results obtained in this paper are an extension and refinement of previously known results. A new random iterative scheme for approximating random common fixed points of three random non-expansive multivalued random operators is defined and we have proved weak and strong convergence theorems in a uniformly convex Banach space.