transition matrix

**davidbooth** · April 1st 2009, 03:32 PM

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?

**Plato** · April 1st 2009, 03:48 PM

$\begin{array}{*{20}c} {} &\vline & A & B & C \\ \hline A &\vline & 0 & 1 & 0 \\ B &\vline & {.75} & 0 & {.25} \\ C &\vline & {.75} & {.25} & 0 \\ \end{array}$

**davidbooth** · April 1st 2009, 03:50 PM

then would i just solve like it's x y z for the a b c?

**Plato** · April 1st 2009, 06:22 PM

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.
$T = \left( {\begin{array}{rrr} 0 & 1 & 0 \\ {.75} & 0 & {.25} \\ {.75} & {.25} & 0 \\ \end{array} } \right)\;\& \;T^4 = \left( {\begin{array}{lll} {.609} & {.188} & {.203} \\ {.293} & {.660} & {.047} \\ {.293} & {.656} & {.051} \\ \end{array} } \right)$

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.

**Berwick** · April 24th 2009, 12:36 AM

Originally Posted by Plato

$\begin{array}{*{20}c} {} &\vline & A & B & C \\ \hline A &\vline & 0 & 1 & 0 \\ B &\vline & {.75} & 0 & {.25} \\ C &\vline & {.75} & {.25} & 0 \\ \end{array}$

Hi Plato, How did you work out the initial matrix???

Thanks.

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