# Math Help - Transition Matrix question

1. ## Transition Matrix question

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?

for a. i got
T=
[0 1 0]
|.6 0 .2|
[.6 .2 0]

im not sure how to do the second part or if i got the right transition matrix for part a. help?

2. I think the matrix should be:

$\left( \begin{array}{ccc}
0 & 1 & 0 \\
& \\
0,75 & 0 & 0,25 \\
& \\
0,75 & 0,25 & 0 \end{array}\right)$

Isn't then an initial-state matrix needed to calculate the rest?

Anyway, if you have an initial state matrix, called $S_0$ and then the transition matrix above, called $P$, the first state matrix, $S_1$ would be calculated as follows: $S_1=S_0P$. The second state matrix as: $S_2=S_0P^2$. The k-th state matrix: $S_k=S_0P^k$

I'm not quite sure about those Markov chains, but I hope that helps.

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?

for a. i got
T=
[0 1 0]
|.6 0 .2|
[.6 .2 0]

im not sure how to do the second part or if i got the right transition matrix for part a. help?