What is the probability that mouse with reach state A before state B

Question

I have a maze as shown above(use the link) and state 3 contains prize while state 7 contains shock. a mouse can be placed in any state from 1 to 9 randomly and it move through the maze uniformly at random

$P_i$ denote the probability that mouse reaches state 3 before state 7, given that AIM started in compartment i.

how to compute $P_i$ for $i ∈ \{1,2,3,4,5,6,7,8,9\}$.

Answer 1

You can use general Markov chain tools for this, as the other answer suggests -- but you can also just wing it:

• Starting in 1, 5, or 9, you can get the answer immediately by symmetry considerations.
• Starting in 3 or 7, the game is over before it starts, so the probability is 100% of one, respective the other.
• Finally, if you start in 2, 4, 6, or 8, after one step you will be in a state that you now already know the probability for. So you just need an appropriately weighted average of those probabilities.

Answer 2

I would write a set of linear equation linking the $P_i$s together and use the fact that $P_3 = 1$ and $P_7 = 0$