Abstract Weighted voting games model decision-making bodies where decisions are made by a majority vote. In such games, each agent has a weight, and a coalition of agents wins the game if the sum of the weights of its members exceeds a certain quota. The Shapley value is used as an index for the true power held by the agents in such games. Earlier work has studied the implications of setting the value of the quota on the agents’ power under the assumption that the game is given with a fixed set of agent weights. We focus on a model where the agent weights originate from a stochastic process, resulting in weight uncertainty. We analyze the expected effect of the quota on voting power given the weight generating process. We examine two extreme cases of the balls and bins model: uniform and exponentially decaying probabilities. We show that the choice of a quota may have a large influence on the power disparity of the agents, even when the governing distribution is likely to result in highly similar weights for the agents. We characterize various interesting repetitive fluctuation patterns in agents’ power as a function of the quota.