In conventional computing systems, the device imperfections constitute the main hindrance on the commercialization of emerging technologies. On the other hand, in alternative computing paradigms, generally acknowledged as unconventional computing, device imperfections can be utilized to achieve complex behaviors that are computationally hard for conventional computers. In Probabilistic Cellular Automata (PCA), complex collective phenomena emerge through simplistic locally coupled probabilistic entities, named as cells. However, the hardware implementations of PCA are highly imposed by the required randomness generation within each PCA cell. In this paper, a novel hardware design of 1-D PCA with Memristors is proposed, utilizing device's non-volatile storage and its unprecedented voltage-controlled probabilistic switching behavior. The necessary theoretical framework for memristor-based PCA (MemPCA) is defined. Moreover, the randomness of MemPCA for various switching probability levels is evaluated through the entropy of generated sequences and the collective effect to all 1-D elementary CA rules is presented, highlighting the effectiveness of memristor as a source of entropy.