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As models in various fields are becoming more complex, associated computational demands have been increasing significantly. Reliability analysis for these systems when failure probabilities are small is significantly challenging, requiring a large number of costly simulations. To address this challenge, this paper introduces Reliability analysis through Error rate-based Adaptive Kriging (REAK). An extension of the Central Limit Theorem based on Lindeberg condition is adopted here to derive the distribution of the number of design samples with wrong sign estimate and subsequently determine the maximum error rate for failure probability estimates. This error rate enables optimal establishment of effective sampling regions at each stage of an adaptive scheme for strategic generation of design samples. Moreover, it facilitates setting a target accuracy for failure probability estimation, which is used as stopping criterion for reliability analysis. These capabilities together can significantly reduce the number of calls to sophisticated, computationally demanding models. The application of REAK for four examples with varying extent of nonlinearity and dimension is presented. Results indicate that REAK is able to reduce the computational demand by as high as 50% compared to state-of-the-art methods of Adaptive Kriging with Monte Carlo Simulation (AK-MCS) and Improved Sequential Kriging Reliability Analysis (ISKRA).