Chaos Game (Sierpinski Triangle)

How does perfect geometric order emerge from pure randomness? The Chaos Game demonstrates this beautifully.
1. Place 3 anchor points (A) forming a triangle.
2. Pick any starting point.
3. Randomly choose one of the 3 anchors, and move halfway towards it. Plot a point.
4. Repeat forever.
Even though every move is completely random, the resulting structure is never random. It always perfectly constructs the Sierpinski Triangle fractal.