The purpose of this paper is to analyze the influence of the random noise on pattern formations in Turing systems with growing domains. Many works in analysis of pattern formations have been performed in the past. However, models in such analyses were considered in a deterministic framework and in time-invariant domains. In this paper, focusing our attention on pattern formations in Turing systems with growing domains and taking disturbances into consideration, we propose a stochastic model of Turing systems with growing domains. We study the influence of the growing domains and random disturbances on Turing pattern formations by numerical simulations.