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Math Help - How would I go about solving this Problem (Matrices)

  1. #1
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    How would I go about solving this Problem (Matrices)

    http://img257.imageshack.us/img257/1311/58856961.png

    I can't seem to figure out how to do it, all I know is I need to use matrices.
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  2. #2
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    Hello, WarZebra!

    This is a Markov process.
    Are your familiar with it?



    . . . . . . . . \begin{array}{cccccccc}C && R && \;S \end{array}
    . . \begin{array}{c}C \\ R \\ S\end{array}\begin{pmatrix}<br />
0.60 & 0.10 & 0.30 \\ 0.20 & 0.55 & 0.25 \\ 0.10 & 0.15 & 0.75 \end{pmatrix}


    (a) If today is cloudy, find the probability that it will be sunny two days from now.

    (b) What about cloudy?

    (c) What about rainy?

    For the two-day probabilities, square the matrix.

    A^2 \;=\;\begin{pmatrix}0.60 & 0.10 & 0.30 \\ 0.20 & 0.55 & 0.25 \\ 0.10 & 0.15 & 0.75\end{pmatrix}\begin{pmatrix}0.60 & 0.10 & 0.30 \\ 0.20 & 0.55 & 0.25 \\ 0.10 & 0.15 & 0.75\end{pmatrix}

    . . . . . . . . . . . \begin{array}{ccccccccccc}C && \;\;R && \quad S \end{array}
    . . . =\; \begin{array}{c}C \\ R \\ S\end{array}\begin{pmatrix}<br />
0.410 & 0.160 & 0.430 \\ 0.255 & 0.3605 & 0.385 \\ 0.165 & 0.205 & 0.630 \end{pmatrix}


    If it is Cloudy today:

    (a)\;P(\text{S in 2 days}) \:=\:0.430

    (b)\;P(\text{C in 2 days}) \:=\:0.410

    (c)\;P(\text{R in 2 days}) \:=\:0.160




    Now do the same calculations for 8 days from now, 10 days from now.
    What do you notice?

    Ridiculous question . . .

    We need to determine \,A^8 and A^{10}.

    And I don't know anyone willing to do that.
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