Critical engineering systems generally demand high reliability while considering uncertainties, which makes failure event identification and reliability analysis based on computer simulation codes computationally very expensive. Rare event can be defined as a failure event that has a small probability of failure value, normally less than 10(-5). This paper presents a new approach, referred to as Accelerated Failure Identification Sampling (AFIS), for probability analysis of rare events, enabling savings of vast computational efforts. To efficiently identify rare failure sample points in probability analysis, the proposed AFIS technique will first construct a Gaussian process (GP) model for system performance of interest and then utilize the developed model to predict unknown responses of Monte Carlo sample points. Second, a new quantitative measure, namely "failure potential", is developed to iteratively search sample points that have the best chance to be a failure sample point. Third, the identified sample points with highest failure potentials are evaluated for the true performance and then used to update the GP model. The failure identification process will be iteratively preceded and the Monte Carlo simulation will then be employed to estimate probabilities of rare events if the maximum failure potential of existing Monte Carlo samples falls below a given target value. Two case studies are used to demonstrate the effectiveness of the developed AFIS approach for rare events identification and probability analysis.