Superfluid turbulence generated by heat flow in He II provides an important physical system for the study of noise-induced transitions. The turbulence is described by a single amplitude, the vortex line density L(\dotQ), and undergoes a continuous transition from a state TI to the homogeneous state TII as the heat current \dotQ is increased. When noise is added to the heat current the system is described by a probability density with maxima defining the most probable state. At low noise levels these stochastic steady states are identical to the deterministic steady states in the absence of noise. This intuitive result is dramatically changed when the noise amplitude is increased. The probability densities are now bimodal in the vicinity of the TI/TII transition and the stochastic steady states show noise-induced bistability.