Resident Evil: The Markov Chain

Yesterday was the first time I taught on the course of Stochastic Processes. No kidding I have been dreaming about it like for 10 years now, and it was awesome. I came up with the following problem to which I don't have the whole solution yet, but as the term passes I'll make the students solve it for me and I'll eventually post the solution. Here it is!

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Jill Valentine (who’s going to be the hero of our story because it’s about to be Valentine’s Day...) is forced to enter a mansion in the middle of the forest because outside the zombie apocalypse is going wild. However, the rooms in the mansion are also filled with zombies and she is compelled to be moving equally likely to any adjacent room to the one she’s in at any given moment in time.

What she doesn’t know is that there’s a room where Nemesis (the code name of a terrible bio-weapon...) is dormant until a member of S.T.A.R.S (Special Tactics and Rescue Squad, and yes, of course Jill is a member!) appears and it will kill her, no doubt about it.

Fortunately, there’s also a helipad (and obviously she can fly the chopper there...) with a helicopter fully functional there. Were she able to reach this room she would fly a way to a secure place after blowing the mansion up with all zombies inside it.

We have a map of the mansion:

1. The rooms in which Jill is in the mansion are a Markov chain. What is the state space? What is the initial distribution? What is the transition probability matrix?

2. What is more likely? Will our heroine survive or will she perish in a not very pleasant way at the hands of the bio-weapon?

3. How long are we expecting to see Jill in action before she reaches her destiny?
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If you solve it before I do, do let me know at my Twitter account :P

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