The main purpose of this paper is to estimate the probability of casting a decisive vote under the majority rule for a class of random electorate models encompassing the celebrated Impartial Culture (IC) and Impartial Anonymous Culture (IAC) models. The emphasis is on the impact of correlation across votes on the order of magnitude of this event. Our proof techniques use arguments from probability theory on one hand and combinatorial and algorithmic tools for counting integer points inside convex polytopes on the other hand.