The work aims to study the first-passage failure probability of single-degree-of-freedom (SDOF) linear and nonlinear package displacement response. For the linear SDOF package, the displacement first-passage problem was analyzed by the geometry in the space of standard normal random variables based on the theoretical analysis. The analytical method for determining the design points was established. The first-passage failure probability was analyzed by the first order reliability method. The relationship between design point excitation and mirror image excitation of the system was analyzed. The method used to determine the design points of nonlinear SDOF package with mirror image excitation was analyzed by the method of optimization. The design point excitations of linear and nonlinear SDOF packages were identical to the mirror image excitation when the system's initial displacement was the threshold. In the steady-state response phase of the system, the design points of the system can be accurately analyzed by mirror image excitation, based on which the displacement first-passage failure probability of the package is analyzed.