In this paper we study the state-feedback stabilization of a discrete-time Markov jump linear system when the observation of the Markov chain of the system, called the Markov state, is time-randomized by another Markov chain. Embedding the Markov state into an extended Markov chain, we transform the given system with time-randomized observations to another one having the enlarged Markov state space but with so-called cluster observations of Markov states. Based on this transformation we propose linear matrix inequalities for designing stabilizing state-feedback gains for the original Markov jump linear systems. The proposed method can treat both periodic observations and many of renewaltype observations in a unified manner, which are studied in the literature using different approaches. A numerical example is provided to demonstrate the obtained result.