Thread: 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?

$\displaystyle \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} $

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

Here is some guidance on this question. $\displaystyle T$ is the transition matrix.
$\displaystyle T^4$ tells us what happens in the fourth week after any given week.
$\displaystyle 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.

Originally Posted by Plato

$\displaystyle \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.