Traversing a Cube

Probability Level 4

A bug starts on one vertex of a cube. Every second he randomly walks along one edge to another vertex. What is the expected value of the number of seconds it will take for him to reach the vertex opposite to the original vertex he was on?

Clarification : Every second he chooses randomly between the three edges available to him, including the one he might have just walked along.

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4 8 6 14 16 10 18 12

1 solution

Geoff Pilling
Oct 4, 2016

The solution is given by this set of linear equations:

$E = 1 + E_1$
$E_1 = 1+ (1/3)*E + (2/3)*E_2$
$E_2 = 1 + (2/3)*E_1$

Where $E$ is the expectation value. $E_1$ is the expectation value from a vertex neighboring the original vertex. $E_2$ is the expectation value from a vertex neighboring the destination vertex.

Solving, $E=\boxed{10}$

Thanks to @Siva Bathula and @Brian Charlesworth for pointing out the issue with the original problem.

Traversing a Cube

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