Expected length of time until the mouse reaches the compost heap.

markov chainsmarkov-processprobabilitystochastic-processes

I'm assuming it's a Markov Chain question but I have no idea how to do this. Thanks in advance!

A mouse lives in a mousehole with three exits.

  • The first exit leads
    to a compost heap after 1 minute of scurrying.
  • The second exit leads
    to a tunnel that returns it to the mousehole after three minutes.
  • The third exit leads to a tunnel that returns it to the mousehole after four
    minutes.

Assume the mouse has no memory of previous excursions
when it returns to its mousehole, and is always equally likely to choose
any one of the exits, what is the expected length of time until the
mouse reaches the compost heap?

The scan of the original question.

Best Answer

Hint: Let $E$ be the desired expected time. Can you explain why $$E = \frac{1}{3}\cdot 1 + \frac{1}{3}(3+E) + \frac{1}{3}(4+E),$$ and hence deduce the value of $E$?

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