This paper proposes a framework to calculate the bounds on failure probability of linear structural systems whose performance is affected by both random variables and interval variables. This kind of problems is known to be very challenging, as it demands coping with aleatoric and epistemic uncertainty explicitly. Inspired by the framework of the operator norm theorem, it is proposed to consider the maximum standard deviation of the structural response as a proxy for detecting the crisp values of the interval parameters, which yield the bounds of the failure probability. The scope of application of the proposed approach comprises linear structural systems, whose properties may be affected by both aleatoric and epistemic uncertainty and that are subjected to (possibly imprecise) Gaussian loading. Numerical examples indicate that the application of such proxy leads to substantial numerical advantages when compared to a traditional double-loop approach for coping with imprecise failure probabilities. In fact, the proposed framework allows to decouple the propagation of aleatoric and epistemic uncertainty.