Hi maryam
The first line of the solution on page 47 is:
m0 = 1 + (2/3)m0 + (1/3)m1
This is the same as:
m0 = (1 + m1) x (1/3) + (1 + m0) x (2/3)
You obtain this by looking at the first time step. You're currently in state 0, with expected time to reach state 2 equal to m0.
What can happen in the first time step? Either moves to state 1 (prob 1/3) or stays in state 0 (prob 2/3).
In the first case, the number of time steps that will be taken will be (1 + m1). This is because the first step has already happened, and now you are in state 1 there are a further m1 steps expected before finally reaching state 2 for the first time.
In the second case, the number of steps taken will be (1 + m0), because one step has already happened, and there are a further m0 steps expected until you can reach state 2 from here. Multiplying by the probabilities (so as to get an expectation) gets you to the above result.
You then simplify this, so as to get the first line of the solution. You can get the second line of the solution on page 47 by the same process but starting in state 1, rather than in state 0. Try it!
Does that help enough?
Robert
