We have calculated the reaction probability and the reaction cross-section of the $N(^4S)+O_2(X^3sum_{g}^{-}); ightarrow;NO(X^2Pi)+O(^3P)$ reaction by the quasiclassical trajectory method with the 6th-order explicit symplectic algorithm, based on a new ground potential energy surface. The advantage of the 6th-order explicit symplectic algorithm, conserving both the total energy and the total angular momentum of the reaction system during the numerical integration of canonical equations, has firstly analyzed in this work, which make the calculation of the reaction probability more reliable. The variation of the reaction probability with the impact parameter and the influence of the relative translational energy on the reaction cross-section of the reaction have been discussed in detail. And the fact is found by the comparison that the reaction probability and the reaction cross-section of the reaction estimated in this work are more reasonable than the theoretical ones determined by Gilibert et al.