Probability, knight on a 5x5 chess board, expected return time.

[Gregg]

Homework Statement

I have a knight on a chess board that is 5x5. I have numbered each position on the board by the amount of steps it takes from that position to get back to the centre.

It looks roughly like this

##\begin{array}{ccccc}
4 & 1 & 2 & 1 & 4 \\
1 & 2 & 3 & 2 & 1 \\
2 & 3 & 0 & 3 & 2 \\
1 & 2 & 3 & 2 & 1 \\
4 & 1 & 2 & 1 & 4
\end{array}##I have made this into a random walk Markov chain with transition matrix.

## \left(
\begin{array}{ccccc}
0 & 1 & 0 & 0 & 0 \\
\frac{1}{3} & 0 & \frac{2}{3} & 0 & 0 \\
0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 \\
0 & 0 & \frac{2}{3} & 0 & \frac{1}{3} \\
0 & 0 & 0 & 1 & 0
\end{array}
\right)##The question is the expected return time starting from the centre.

I have solved the set of equations

## k_i^{\{A\}} = 0, \text{ if } i\in A ##
## k_i^{\{A\}} = 1+\sum_j p_{ij} k_j^{\{A\}}, \text{ if } i\in A^C ##I get

##k_0^{\{0\}} = 0##
##k_1^{\{0\}} = 11##
##k_2^{\{0\}} = 15##
##k_3^{\{0\}} = 17##
##k_4^{\{0\}} = 18##What do I do with these?

Edit: my guess is: Since ##P_{01} = 1 ## the answer is ##11+1=12##

If this is the case, say if there was some probability of hitting all of the other states from 0, would the expected return time be ##k^{\{0\}} = 1 + \sum_{j} p_{ij} k_j^{\{0\}} ## ?

 

[lippyka]

Homework Equations ## k_i^{\{A\}} = 0, \text{ if } i\in A #### k_i^{\{A\}} = 1+\sum_j p_{ij} k_j^{\{A\}}, \text{ if } i\in A^C ##The Attempt at a Solution My guess is: Since ##P_{01} = 1 ## the answer is ##11+1=12##If this is the case, say if there was some probability of hitting all of the other states from 0, would the expected return time be ##k^{\{0\}} = 1 + \sum_{j} p_{ij} k_j^{\{0\}} ## ?