Markov chain problems, absorbing states and eigenvalues Problem set A

Members of a town library choose one of three branches each week: Main (M), Central (C) or South (S). The transition diagram shows the probabilities of changing branch from one week to the next. Write down the transition matrix.

A frog sits on one of three lily pads, A, B or C. Each minute it moves according to the transition matrix \(T=\begin{pmatrix}0.46 & 0.28 & 0 \\ 0.54 & 0.22 & 0.64 \\ 0 & 0.5 & 0.36\end{pmatrix}\) (columns and rows in the order A, B, C). Draw the transition diagram, with the probability on each arrow.

A train is On time (O) or Late (L) each day. If it is on time today, the probability that it is on time tomorrow is 0.85. If it is late today, the probability that it is on time tomorrow is 0.35. Write down the transition matrix, with states in the order O, L.

A hiker spends each night at one of three huts, A, B or C. Where she spends tomorrow night depends only on where she spends tonight, with transition matrix \(T=\begin{pmatrix}0.42 & 0.28 & 0.12 \\ 0.38 & 0.52 & 0.36 \\ 0.2 & 0.2 & 0.52\end{pmatrix}\) (columns and rows in the order A, B, C).

1. (a) She is at hut A on Monday night. Find the probability that she is at hut C on Wednesday night.
2. (b) She is at hut B on Monday night. Find the probability that she is at hut B on Wednesday night.

Each day a commuter travels by Train or by Bus. The transition matrix is \(T=\begin{pmatrix}0.85 & 0.3 \\ 0.15 & 0.7\end{pmatrix}\) (columns and rows in the order Train, Bus). In the long term the proportions of days by train and by bus are \(x\) and \(y\).

1. (a) Write down a system of equations for the long-term proportions.
2. (b) Solve the equations to find the exact long-term proportions.

A phone company has 1000 customers, each on the Basic (B), Standard (S) or Premium (P) plan. Each month, of the customers on Basic, 70% stay and 20% move to Standard (the rest move to Premium). Of the customers on Standard, 10% move to Basic and 60% stay. Of the customers on Premium, none move to Basic and 80% stay. Initially 500 customers are on Basic, 300 on Standard and 200 on Premium.

1. (a) Write down the transition matrix T, with the states in the order B, S, P.
2. (b) Find the number of customers on each plan after 6 months. Give each answer to the nearest whole number.
3. (c) Find the number of customers on each plan in the long term, to the nearest whole number.

A rental firm has three depots: Airport (A), City (C) and Harbour (H). A car is returned each day to one depot, and the depot depends only on where it was returned the day before, with transition matrix \(T=\begin{pmatrix}0.52 & 0.18 & 0.32 \\ 0.28 & 0.62 & 0.18 \\ 0.2 & 0.2 & 0.5\end{pmatrix}\) (columns and rows in the order A, C, H). A car is returned to the Airport today. Find the probability that it is returned to the City depot four days later.

Each week a user of a fitness app is Daily (D), Weekly (W) or Inactive (I). The transition matrix is \(T=\begin{pmatrix}0.68 & 0.28 & 0.12 \\ 0.22 & 0.52 & 0.18 \\ 0.1 & 0.2 & 0.7\end{pmatrix}\) (columns and rows in the order D, W, I). Use technology to find the long-term probability that a user is a Daily user.

A gambler starts a game with \(\pounds 2\). He bets \(\pounds 1\) each round. He wins \(\pounds 1\) with probability 0.45 and loses \(\pounds 1\) with probability 0.55. He stops when he has \(\pounds 0\) or \(\pounds 5\).

1. (a) Write down the transition matrix for the amount of money he has, with states \(\pounds 0\) to \(\pounds 5\).
2. (b) Find the probability that he finishes with \(\pounds 5\).

Customers of a delivery app are Active or Lapsed. Each month 0.3 of active customers lapse and 0.12 of lapsed customers become active. The transition matrix is \(T=\begin{pmatrix}0.7 & 0.12 \\ 0.3 & 0.88\end{pmatrix}\) (columns and rows in the order Active, Lapsed).

1. (a) Find the eigenvalues of T and a corresponding eigenvector for each.
2. (b) Write down matrices P and D such that \(T=PDP^{-1}\).

A lake is Clear or Cloudy each week. The transition matrix is \(T=\begin{pmatrix}0.82 & 0.42 \\ 0.18 & 0.58\end{pmatrix}\) (columns and rows in the order Clear, Cloudy), and \(T=PDP^{-1}\) where \(P=\begin{pmatrix}7 & 1 \\ 3 & -1\end{pmatrix},\ D=\begin{pmatrix}1 & 0 \\ 0 & 0.4\end{pmatrix}\). At the start, the probability that the lake is clear is 0.9. Let \(f_n\) be the probability that the lake is clear after \(n\) weeks.

1. (a) Find an expression for \(f_n\).
2. (b) Hence find the long-term probability that the lake is clear.

A family chooses a holiday each year: Beach (B), Mountains (M) or City (C). The transition matrix is \(T = \begin{pmatrix} 0.5 & 0.25 & 0.05 \\ 0.25 & 0.5 & 0.2 \\ 0.25 & 0.25 & 0.75 \end{pmatrix}\), in that order.

1. (a) Use your calculator to find the three eigenvalues of \(T\).
2. (b) Write down an eigenvector for the eigenvalue 1.
3. (c) Hence find the steady-state vector, as exact fractions.