This paper focuses on probabilistic matching systems where two classes of users arrive at the system to match with users from the other class. The users are selective and the matchings occur probabilistically. Recently, Markov chain models were proposed to analyze these systems; however, an exact analysis of these models to completely characterize the performance is not possible due to the probabilistic matching structure. In this work, we propose approximation methods based on fluid and diffusion limits using different scalings. We analyze the basic properties of these approximations and show that some performance measures are insensitive to the matching probability, agreeing with the existing results. We also perform numerical experiments with our approximations to gain insight into probabilistic matching systems.