We study the probability of multistability in a quadratic map driven repeatedly by a random signal of length N, where N is taken as a measure of the signal complexity. We first establish analytically that the number of coexisting attractors is bounded above by N. We then numerically estimate the probability p of a randomly chosen signal resulting in a multistable response as a function of N. Interestingly, with increasing drive signal complexity the system exhibits a paucity of attractors. That is, almost any drive signal beyond a certain complexity level will result in a single attractor response (p=0). This mechanism may play a role in allowing sensitive multistable systems to respond consistently to external influences.