The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave it. However, the choice of the beginning point is not important! The final shape will be quite like the real triangle, even if the first point is not on the triangle! Why?
P.S. I'm still working on my tomorrow's lecture!