We investigate the stability of Boolean networks (BNs) with impulses triggered by both states and random factors. A hybrid index model is used to describe impulsive BNs. First, several necessary and sufficient conditions for forward completeness are obtained. Second, based on the stability criterion of probabilistic BNs and the forward completeness criterion, the necessary and sufficient conditions for the finite-time stability with probability one and the asymptotical stability in distribution are presented. The relationship between these two kinds of stability is discussed. Last, examples and time-domain simulations are provided to illustrate the obtained results.