Ant on a Cube
Probability
Level
3
An art starts at a corner of a cube. A 'move' is a journey through an edge from one corner to another.
If the ant moves utterly randomly, what is the expected number of moves it will take to reach the entirely opposite corner of the cube (the furthest corner away)?
The answer is 10.
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1 solution
We consider the 'states' of the ant to be $0,1,2,3$ where $n$ is the minimum number of moves away from the finish. So consider $E_n$ as the expected number of moves from state $n$.
We arrive at the following equations: $$E 3=1+E 2$$ $$E 2=\frac{1}{3}(1+E 3)+\frac{2}{3}(1+E 1)$$ $$E 1=\frac{2}{3}(1+E 2)+\frac{1}{3}(1+E 0$$ where $E_0=0$ (no more moves required).
Solving this via elementary row operations or otherwise, gives $E_3=10$.
Thus the expected number of moves