Stochastic games are a natural model for the synthesis of controllers confronted to adversarial and/or random actions. In particular, $\omega$-regular games of infinite length can represent reactive systems which are not expected to reach a correct state, but rather to handle a continuous stream of events. One critical resource in such applications is the memory used by the controller. In this paper, we study the amount of memory that can be saved through the use of randomisation in strategies, and present matching upper and lower bounds for stochastic Muller games.