Reliable and low latency multicast communication is important for future vehicular communication. Sparse random linear network coding approach used to ensure the reliability of multicast communication has been widely investigated. A fundamental problem of such communication is to characterize the decoding success probability, which is given by the probability of a sparse random matrix over a finite field being full rank. However, the exact expression for the probability of a sparse random matrix being full rank is still unknown, and existing approximations are recursive or not consistently tight. In this paper, we provide a tight and closed-form approximation to the probability of a sparse random matrix being full rank, by presenting the explicit structure of the reduced row echelon form of a full rank matrix and using the product theorem. Simulation results show that our proposed approximation is of high accuracy regardless of the generation size, the number of coded packets, the field size and the sparsity, and tighter than the state-of-the-art approximations for a large range of parameters.