In H2G2 Logic: As every object in the universe exerts an influence on every other object in the universe it is in fact possible to extrapolate all of creation from a small fairy cake. Hence it would be a simple extension of the principle used in the total perspective vortex for anyone to know the exact state of your brownian motion producer at any point so long as they have easy access to the necessary baked good.
In the real world: I would say brownian motion is pretty damn random. Theoretically if you new the precise position and velocity of every molecule in the tea at one instant you might be able to predict where it will be at some point in the future. However, the Heisenberg Uncertainty Principle means that it is impossible to do this even if you have access to the other persons cup of tea.