Question 77
Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons, and regular octagons. Tim makes a random sequence $F_{0}, F_{1}, F_{2}, F_{3}, \ldots$ of figures as follows:
- $F_{0}$ is a square.
- For every positive integer $i$, $F_{i}$ is randomly chosen to be one of the 2 figures distinct from $F_{i-1}$ (each chosen with equal probability $\frac{1}{2}$).
- Tim takes 4 seconds to make squares, 6 to make hexagons, and 8 to make octagons. He makes one figure after another, with no breaks in between.
Suppose that exactly 17 seconds after he starts making $F_{0}$, Tim is making a figure with $n$ sides. What is the expected value of $n$?
- $F_{0}$ is a square.
- For every positive integer $i$, $F_{i}$ is randomly chosen to be one of the 2 figures distinct from $F_{i-1}$ (each chosen with equal probability $\frac{1}{2}$).
- Tim takes 4 seconds to make squares, 6 to make hexagons, and 8 to make octagons. He makes one figure after another, with no breaks in between.
Suppose that exactly 17 seconds after he starts making $F_{0}$, Tim is making a figure with $n$ sides. What is the expected value of $n$?