In binary cascade dynamics, the nodes of a graph are in one of two possible states (inactive, active), andnodes in the inactive state make an irreversible transition to the active state, as soon as their precursors satisfya predetermined condition. We introduce a set of recursive equations to compute the probability of reachingany final state, given an initial state, and a specification of the transition probability function of each node.Because the naive recursive approach for solving these equations takes factorial time in the number of nodes, wealso introduce an accelerated algorithm, built around a breath-first search procedure. This algorithm solves theequations as efficiently as possible in exponential time.