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The integration of Reconfigurable Intelligent Surface (RIS) with the Dual-Functional Radar-Communication (DFRC) base station in a unified system has been a topic of growing interest. This study centers on a system augmented by RIS for both target sensing and multi-user communication purposes. Our primary objective is to enhance radar efficacy. To this end, we devise a framework to maximize radar detection probability, taking into account the combined aspects of transmit beamforming and RIS phase-shift vector formulation, ensuring we adhere to set standards for communication quality of service (QoS), transmit power limitations, and phase-shift parameters. Firstly, we obtain the target angle by maximum likelihood estimation. Then, we perform target tracking by solving a Quadratic Semidefinite Programming (QSDP) problem. Next, we maximize the detection probability for different scenarios, i.e., general case …