Quality inconsistency due to uncertainty hinders the extensive applications of a laser powder bed fusion (L-PBF) additive manufacturing process. To address this issue, this study proposes a new and efficient probabilistic method for the reliability analysis and design of the L-PBF process. The method determines a feasible region of the design space for given design requirements at specified reliability levels. If a design point falls into the feasible region, the design requirement will be satisfied with a probability higher or equal to the specified reliability. Since the problem involves the inverse reliability analysis that requires calling the direct reliability analysis repeatedly, directly using Monte Carlo simulation (MCS) is computationally intractable, especially for a high reliability requirement. In this work, a new algorithm is developed to combine MCS and the first-order reliability method (FORM). The algorithm finds the initial feasible region quickly by FORM and then updates it with higher accuracy by MCS. The method is applied to several case studies, where the normalized enthalpy criterion is used as a design requirement. The feasible regions of the normalized enthalpy criterion are obtained as contours with respect to the laser power and laser scan speed at different reliability levels, accounting for uncertainty in seven processing and material parameters. The results show that the proposed method dramatically alleviates the computational cost while maintaining high accuracy. This work provides a guidance for the process design with required reliability.