Powers of the transition matrix and state vectors Problem set B

Each matrix has the states in the order A, B. Use your calculator.

1. (a) \(\mathbf{T} = \begin{pmatrix} 0.62 & 0.17 \\ 0.38 & 0.83 \end{pmatrix}\). Find the probability of being in state B after 4 steps, given that you start in A.
2. (b) \(\mathbf{T} = \begin{pmatrix} 0.13 & 0.52 \\ 0.87 & 0.48 \end{pmatrix}\). Find the probability of being in state A after 3 steps, given that you start in B.

\(\mathbf{T} = \begin{pmatrix} 0.57 & 0.24 & 0.09 \\ 0.12 & 0.49 & 0.33 \\ 0.31 & 0.27 & 0.58 \end{pmatrix}\), with the states in the order A, B, C. Find the probability of being in state A after 5 steps, given that you start in C.

\(\mathbf{T} = \begin{pmatrix} 1 & 0.17 & 0.08 & 0.06 \\ 0 & 0.43 & 0.14 & 0.22 \\ 0 & 0.27 & 0.52 & 0.12 \\ 0 & 0.13 & 0.26 & 0.6 \end{pmatrix}\), with the states in the order A, B, C, D. State A is absorbing.

1. (a) Find the probability of being in state A after 6 steps, given that you start in D.
2. (b) Find the probability of being in state D after 3 steps, given that you start in B.

\(\mathbf{T} = \begin{pmatrix} 0.39 & 0.72 \\ 0.61 & 0.28 \end{pmatrix}\) and \(\mathbf{s}_0 = \begin{pmatrix} 0.2 \\ 0.8 \end{pmatrix}\). Find \(\mathbf{s}_4\).

\(\mathbf{T} = \begin{pmatrix} 0.83 & 0.06 & 0.11 \\ 0.04 & 0.79 & 0.18 \\ 0.13 & 0.15 & 0.71 \end{pmatrix}\) and \(\mathbf{s}_0 = \begin{pmatrix} 0.25 \\ 0.35 \\ 0.4 \end{pmatrix}\). Find \(\mathbf{s}_3\).

For each transition matrix, say whether the chain is regular. If it is, find the smallest power of the matrix with every entry positive. If it is not, say why.

1. (a) \(\mathbf{T} = \begin{pmatrix} 0.7 & 1 \\ 0.3 & 0 \end{pmatrix}\), with two states.
2. (b) \(\mathbf{T} = \begin{pmatrix} 0.6 & 0 & 0 \\ 0.1 & 1 & 0.3 \\ 0.3 & 0 & 0.7 \end{pmatrix}\), with three states.
3. (c) \(\mathbf{T} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), with two states.

Three mobile networks, Nova, Orbit and Pulse, have market shares of 25%, 50% and 25%. Each quarter customers move between the networks with transition matrix \(\mathbf{T} = \begin{pmatrix} 0.78 & 0.06 & 0.14 \\ 0.12 & 0.88 & 0.08 \\ 0.1 & 0.06 & 0.78 \end{pmatrix}\), in that order. Find the projected market share of each network after 5 quarters.

A lift stops at three floors, A, B and C, at the end of each trip. The next floor depends on the present floor, with transition matrix \(\mathbf{T} = \begin{pmatrix} 0.57 & 0.18 & 0.14 \\ 0.31 & 0.63 & 0.26 \\ 0.12 & 0.19 & 0.6 \end{pmatrix}\), in that order. The lift is at floor A. Find the probability that it is at floor C after 4 more trips.

A shop has 1200 loyalty customers. At the start, 300 use the app (P) and 900 use the card (C). Each week the customers change method with transition matrix \(\mathbf{T} = \begin{pmatrix} 0.82 & 0.43 \\ 0.18 & 0.57 \end{pmatrix}\), in the order P, C. Find the number of customers using each method after 4 weeks.

A morning bus is On time (O) or Late (L) each day. The transition matrix is \(\mathbf{T} = \begin{pmatrix} 0.7 & 0.45 \\ 0.3 & 0.55 \end{pmatrix}\), with the states in the order O, L. The bus is on time on Monday.

1. (a) Find the probability that Tuesday and Wednesday are both late and Thursday is on time.
2. (b) Find the probability that Thursday is on time, whatever happens on Tuesday and Wednesday.

Three bus routes P, Q and R carry 70%, 20% and 10% of the passengers. Each year the shares change with transition matrix \(\mathbf{T} = \begin{pmatrix} 0.72 & 0.12 & 0.06 \\ 0.21 & 0.78 & 0.11 \\ 0.07 & 0.1 & 0.83 \end{pmatrix}\), in that order. Find the first year in which route R carries more than 30% of the passengers.

\(\mathbf{T} = \begin{pmatrix} 0.37 & 0.28 & 0.12 \\ 0.41 & 0.32 & 0.52 \\ 0.22 & 0.4 & 0.36 \end{pmatrix}\), with the states in the order A, B, C.

1. (a) Find the probability of going from B to A in exactly two steps by adding the probabilities of the three routes.
2. (b) Check your answer by finding \(\mathbf{T}^2\) on your calculator.