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Here’s the problem we’ll be tackling:

An ant is at a vertex of a tetrahedron. Every second, it randomly chooses one of the other 3 vertices, and crawls to that vertex. What is the expected number of seconds until it has visited every vertex?

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Let’s define some variables to help solve this problem. Which do you think would be the most helpful?

**A:**\(X_i,\) the expected number of distinct vertices visited after \(i\) seconds**B:**\(X_i,\) the expected number of seconds until \(i\) distinct vertices have been visited

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If \(X_i\) is the expected number of seconds until \(i\) distinct vertices have been visited, what is \(X_1?\)

Remember that the ant starts at a vertex (at time 0).

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Using the same logic, you can determine the expected number of seconds between visiting the third vertex and final vertex.

Using this and all prior knowledge, what is \(X_4,\) the amount of total expected time in seconds from when the ant starts moving at 0 seconds?

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