IB Maths AI HL AI 4.5 Markov chains questions

A campus club records which of two cafés its members visit each Friday. The transition diagram in Figure 1 gives the probabilities that a member visits each café on the next Friday, based on the café visited this Friday.

Let $s_n$ be the column state matrix after $n$ weekly transitions, using the state order North, South. On the Friday of the survey, the state matrix is

$$s_0=\begin{pmatrix}0.75\\0.25\end{pmatrix}.$$

A rail authority models the monthly change in commuters’ usual station between North (N) and South (S). The state matrix is a column matrix, with entries giving the proportions using N and S, in that order. The transition matrix is

$$T=\begin{pmatrix}0.8&0.3\\0.2&0.7\end{pmatrix},$$

where rows represent the station used in the next month and columns represent the station used in the current month. Initially,

$$s_0=\begin{pmatrix}0.10\\0.90\end{pmatrix}.$$

A university models changes in students’ preferred meal-collection hub (A, B or C) from one month to the next. The columns of the transition matrix $T$ represent the current hub, and the rows represent the hub in the following month. The state matrices are column matrices.

$$T= \begin{pmatrix} 0.6&0.2&0.1\\ x&0.5&0.2\\ 0.4-x&0.3&0.7 \end{pmatrix}$$

The initial state matrix and the state matrix one month later are

$$s_0= \begin{pmatrix} 0.50\\ 0.30\\ 0.20 \end{pmatrix}, \qquad s_1= \begin{pmatrix} 0.38\\ 0.34\\ 0.28 \end{pmatrix}.$$

The same transition matrix applies each month, and the resulting Markov chain is regular.