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Random Ant
An ant starts at a specific corner of a cube and tries to reach the exact opposite corner of the cube. It walks randomly directly to neighbor corners with an equal probability, 1/3. Every time the ant travels to the next corner, a second passes. What is the expected number of seconds that it takes the ant to reach the exact opposite corner?
Figure 1 - Imagination of the cube.
Hint
Look at the given figure in the question. Number the corners and write out the Markov chain.
Solution
First, start with numbering all corners to get a clear overview of all states. The ant starts at 
and wants to reach 
. This problem can be approached as a Markov chain problem. We can start writing out all state equations:
is equal to zero, because it is the absorption state. We can now substitute 
in Equation 2. This gives us: 
in in Equation 1:
and wants to reach . This problem can be approached as a Markov chain problem. We can start writing out all state equations:
(1) 
(2) 
(3) 
is equal to zero, because it is the absorption state. We can now substitute in Equation 2. This gives us: (4) 
(5) 
(6) 
in in Equation 1: (7) 
(8) 
(9) 
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