This paper discusses robustly stabilizing state feedback synthesis of discrete-time stochastic plants whose dynamics are characterized by convex polytopes (called stochastic polytopes) consisting of random matrices (i.e., matrices involving random variables). The stochastic polytopes enable us to describe the uncertainties in the probability distributions underlying the stochastic systems. Hence, we can study robust stability (in the stochastic sense) of the systems with respect to the uncertainties in the distributions, through dealing with stochastic polytopes. This paper gives a synthesis-oriented sufficient condition for robust closed-loop stability, and states a numerical design method exploiting the condition. The effectiveness of the method is also demonstrated with a numerical example.