Markov chain average waiting time

Question

Given the following $E=\{1,2,3\}$ and matrix $P=\begin{bmatrix} 1/2 & 1/3 & 1/6\\ 1/4 & 3/4 & 0\\ 1/2& 0 & 1/2 \end{bmatrix}$, assuming that chain starts from point 1 find average time required to reach state two. I know i have to set up a system of equations: $$ t_2=0, t_1=1/2t_1+1/3t_2+1/6 t_3+1, t_3=1/2t_3+1/2t_1+1 $$ where t stands for time. However the only thing i do not understand is this 1 we are adding. Why is that?

Answer 1

The idea of the equations is conditioning on the first step. In words, the mean time to hit state 2 starting from state 1 is the sum of the mean time to hit state 2 starting from state i, times the probability to go directly to state i from state 1, plus one, to account for the first step itself.

In symbols you might write something like:

$$E[\tau \mid X_0=1]=\sum_i E[\tau \mid X_0=1,X_1=i]P[X_1=i \mid X_0=1] \\=\sum_i (E[\tau \mid X_0=i]+1)P[X_1=i \mid X_0=1]$$

where we have used the Markov property to simplify the conditional expectations in the sum.