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

  1. May 15, 2012 #1
    1. The problem statement, all variables and given/known data

    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\}} ## ?
     
    Last edited: May 15, 2012
  2. jcsd
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