One-point quadrature discretization is one of the most popular discretizations for peridynamic simulation, but it suffers from poor accuracy. Employing smoothly decaying influence functions or precise volume correction can address this, but the former applies only to specific materials, and the latter may be computationally expensive and limited to square or cubic meshes. This paper discusses one-point quadrature discretization in the context of Monte Carlo integration and presents the quadrature rule, considering the smoothed acceptance-rejection method. In this context, the volume correction factor is explained as the probability of two particles interacting with each other. We summarize several volume correction algorithms from the literature and then present a new one based on the proposed quadrature rule. Numerical tests were conducted in both 2D and 3D. According to the comparison, the proposed algorithm only costs approximately 0.2% to 40% of the time that precise volume correction needs to compute the quadrature weights, while its integration accuracy is still at the same level. The results suggest that the continuity of the acceptance-rejection rule and the precision of the expectation of the horizon volume primarily matter in the accuracy of quadrature rules.