This paper addresses the common engineering practice of specifying a required probability of attaining some performance level. The problem setup is that of a standard robust H_∞ performance analysis of a parameter-dependent system, except that the parameter hyper-rectangle shrinks in the analysis in order to accommodate a performance goal that is better than the one attainable for the original parameter box. An affine-quadratic, multiconvex approach is applied to reduce the overdesign that is inherent in the quadratic approach. A version of the Bounded Real Lemma (BRL) in a form of Bi-Linear Matrix Inequalities (BLMIs) guarantees a minimum H_∞-norm for a prescribed probability. These BLMIs are solved using an iterative algorithm. A uniform distribution is assumed for the system parameters, according to the uniformity principle. The probability requirement is expressed by a set of LMIs that is derived by an extension of an existing second-order cone method. The latter LMIs are to be concurrently solved with the BLMIs of the BRL. The features of the proposed analysis are demonstrated via a two-parameter example.