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Math Help - transition matrix

  1. #1
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    transition matrix

    A housekeeper buys three kinds of cereal: A, B, C. She never buys the same cereal in successive weeks. If she buys cereal A, then the next week she buys cereal B. However, if she buys either B or C, then the next week she is three times as likely to buy A as the other brand.
    (a) Find the transition matrix.
    (b) In the long run, how often does she buy each of the three brands?

    How do I go about a matrix without numbers?
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  2. #2
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    \begin{array}{*{20}c}<br />
   {} &\vline &  A & B & C  \\<br />
\hline<br />
   A &\vline &  0 & 1 & 0  \\<br />
   B &\vline &  {.75} & 0 & {.25}  \\<br />
   C &\vline &  {.75} & {.25} & 0  \\<br /> <br />
 \end{array}
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  3. #3
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    then would i just solve like it's x y z for the a b c?
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  4. #4
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    Here is some guidance on this question. T is the transition matrix.
    T^4 tells us what happens in the fourth week after any given week.
    T = \left( {\begin{array}{rrr}<br />
   0 & 1 & 0  \\<br />
   {.75} & 0 & {.25}  \\<br />
   {.75} & {.25} & 0  \\<br /> <br />
 \end{array} } \right)\;\& \;T^4  = \left( {\begin{array}{lll}<br />
   {.609} & {.188} & {.203}  \\<br />
   {.293} & {.660} & {.047}  \\<br />
   {.293} & {.656} & {.051}  \\ \end{array} } \right)

    Suppose in week zero the housekeeper bought brand A in the fourth week after that the probability that she/he buys A again is .609, on the other hand the probability that she/he buys brand C is .203.

    Suppose in week zero the housekeeper bought brand C in the fourth week after that the probability that she/he buys B is .656, on the other hand the probability that she/he buys brand C again is .051.
    Last edited by Plato; April 1st 2009 at 06:35 PM.
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  5. #5
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    Quote Originally Posted by Plato View Post
    \begin{array}{*{20}c}<br />
   {} &\vline &  A & B & C  \\<br />
\hline<br />
   A &\vline &  0 & 1 & 0  \\<br />
   B &\vline &  {.75} & 0 & {.25}  \\<br />
   C &\vline &  {.75} & {.25} & 0  \\<br /> <br />
 \end{array}
    Hi Plato, How did you work out the initial matrix???

    Thanks.
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