Fuzzy probability offers a framework for taking into account the effects of both aleatoric and epistemic uncertainty on the performance of a system, quantifying its level of safety, for example, in terms of a fuzzy failure probability. However, the practical application of fuzzy probability is often challenging due to increased numerical efforts arising from the need to propagate both types of uncertainties. Hence, this contribution proposes an approach for approximate calculation of fuzzy failure probabilities for a class of problems that involve moderately nonlinear performance functions, where uncertain input parameters of a model are characterized as random variables while their associated distribution parameters (for example, mean and standard deviation) are described as fuzzy variables. The proposed approach is cast as a post-processing step of a standard (yet advanced) reliability analysis. The key issue for performing an approximate calculation of the fuzzy failure probabilities is extracting probability sensitivity information from the reliability analysis stage as well as the introduction of intervening variables that capture - to some extent - the nonlinear relation between distribution parameters and the failure probability. A series of relatively simple illustrative examples demonstrate the capabilities of the proposed approach, highlighting its numerical advantages, as it comprises a single standard reliability analysis plus some additional system analyses.