Feedback control of a discrete-time switched stochastic system is explored for the case where the active operation mode is observed only at random time instants. A stabilizing control law that utilizes the information obtained through mode observations is proposed. We analyze the probabilistic dynamics of a sequence-valued stochastic process that captures the key properties of the evolution of active mode between mode observation instants. We then use the results of our analysis to obtain sufficient conditions under which our proposed control law guarantees almost sure asymptotic stabilization. We illustrate the efficacy of our approach with a numerical example.