Markov Chains & Transition Matrices: exam review Exam review: second paper

A runner plans each day to Run (R), Swim (S) or take a rest day (T). After a run, she swims the next day with probability 0.35 and rests with probability 0.25. After a swim, she runs the next day with probability 0.3 and swims again with probability 0.5. After a rest day, she never swims the next day, and she runs with probability 0.55. [6]

1. (a) (a) Write down the transition matrix, with the states in the order R, S, T. [4]
2. (b) (b) Draw the transition diagram. [2]

A phone contract is Active (A), Paused (P) or Cancelled (X) at the start of each month. The transition diagram is shown. [7]

1. (a) (a) Write down the transition matrix, with the states in the order A, P, X. [3]
2. (b) (b) Write down the absorbing state and say how the matrix shows it. [1]
3. (c) (c) A contract is active this month. Find the probability that the contract is cancelled after 4 months. [3]

Each day in a coastal town is either Wet (W) or Dry (D). The states in the order W, D have transition matrix \(\mathbf{T}=\begin{pmatrix}0.75 & 0.4 \\ 0.25 & 0.6\end{pmatrix}\) [6]

1. (a) (a) Today is dry. Find the probability that it is wet five days from now. [3]
2. (b) (b) On a certain day there is a probability of 0.3 that the day is wet. Find the probability of each state four days later. [3]

A chain of cafes has three branches, P, Q and R. Each month its regular customers choose a branch to visit. The transition matrix, with the branches in the order P, Q, R, is \(\mathbf{T}=\begin{pmatrix}0.6 & 0.15 & 0.2 \\ 0.25 & 0.7 & 0.1 \\ 0.15 & 0.15 & 0.7\end{pmatrix}\) [7]

1. (a) (a) Write down a system of three linear equations that the steady-state vector must satisfy. Use \(\mathbf{T}\mathbf{s}=\mathbf{s}\) and the fact that the shares add to 1. [2]
2. (b) (b) Solve your system to find the exact steady-state vector. [3]
3. (c) (c) The chain has 2400 regular customers. In the long term, how many visit branch P each month? Give your answer to the nearest whole number. [2]

A shop's customers pay each week by Card (C) or by Cash (H). The way a customer pays in one week depends only on the way they paid in the previous week. The transition diagram is shown. [6]

1. (a) (a) Write down the transition matrix, with the states in the order C, H. [2]
2. (b) (b) A customer pays by card in week 1. Find the probability that the customer pays cash in weeks 2 and 3 and by card in week 4. [2]
3. (c) (c) A customer pays by card in week 1. Find the probability that the customer pays cash in week 3. [2]

The users of an app are either Active (A) or Inactive (I) each month. 70% of the users who are active in one month are active the next month. 40% of the users who are inactive in one month are active the next month. There are 500 users. This month 300 of them are active. [8]

1. (a) (a) Write down the transition matrix, with the states in the order A, I. [2]
2. (b) (b) Find the number of users expected to be active in 3 months' time. [2]
3. (c) (c) Use a high power of \(\mathbf{T}\) to find the number of users expected to be active in the long term. [2]
4. (d) (d) Show, by solving \(\mathbf{T}\mathbf{s}=\mathbf{s}\), that the long-term proportion in the first state is exactly \(\frac{4}{7}\). [2]

The water level in a reservoir is recorded each week as High (H), Medium (M) or Low (L). The level next week depends only on the level this week. After a High week, the next week is High with probability 0.6 and Medium with probability 0.3. After a Medium week, it is High with probability 0.25 and Low with probability 0.35. After a Low week, it is High with probability 0.15 and Medium with probability 0.35. [14]

1. (a) (a) Write down the transition matrix, with the states in the order H, M, L. [4]
2. (b) (b) The level is medium this week. Find the probability that it is low three weeks from now. [2]
3. (c) (c) At the start of the study the probabilities of a high, a medium and a low level are 0.2, 0.5 and 0.3. Find the probabilities of each level four weeks later. [3]
4. (d) (d) Find the long-term matrix, giving its entries to 3 significant figures. [2]
5. (e) (e) Write down the long-term percentage of weeks in which the level is low. [1]
6. (f) (f) Explain why every column of the long-term matrix is the same. [2]

A robot moves along a track of five stations, numbered 0 to 4. Each hour it moves one station up with probability 0.4 and one station down with probability 0.6. Stations 0 and 4 are docks: once the robot is in one it stays there. The robot starts at station 3. [14]

1. (a) (a) Write down the two absorbing states and say what absorbing means. [2]
2. (b) (b) Write down the transition matrix, with the stations in the order 0 to 4. [3]
3. (c) (c) Find the probability that the robot is at station 2 after 3 hours. [2]
4. (d) (d) Find the probability that the robot finishes at the dock at station 4. [3]
5. (e) (e) Explain why this chain has no single steady state. [2]
6. (f) (f) 400 robots start at station 3. How many do you expect to finish at the dock at station 4? [2]

Each time a customer upgrades their phone they buy either an Android phone (A) or an iPhone (I). After owning an Android phone, 20% of customers switch to an iPhone at the next upgrade. After owning an iPhone, 30% of customers switch to an Android phone. There are no other makes. [15]

1. (a) (a) Write down the transition matrix \(\mathbf{T}\), with the states in the order A, I. [2]
2. (b) (b) Find the eigenvalues of \(\mathbf{T}\). [3]
3. (c) (c) Find an eigenvector for each eigenvalue. [3]
4. (d) (d) Write down matrices \(\mathbf{P}\) and \(\mathbf{D}\) such that \(\mathbf{T}=\mathbf{PDP}^{-1}\), where \(\mathbf{D}\) is diagonal. [2]
5. (e) (e) At the start, 45% of the customers own an Android phone. Use \(\mathbf{T}^n=\mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}\) to find an expression for the proportion of Android owners after \(n\) upgrades. [3]
6. (f) (f) Hence write down the long-term proportion of Android owners. [2]

A warehouse has four zones, P, Q, R and S. Every hour each parcel stays in its zone or is moved to another zone. The probabilities of moving are known for a parcel in each zone. From P: to Q 0.25, to R 0.05, to S 0.2. From Q: to P 0.1, to R 0.3, to S 0.15. From R: to P 0.2, to Q 0.1, to S 0.3. From S: to P 0.25, to R 0.15, and never to Q. [12]

1. (a) (a) Write down the transition matrix, with the zones in the order P, Q, R, S. [3]
2. (b) (b) A parcel is in zone P. Find the probability that it is in zone S four hours later. [2]
3. (c) (c) At 08:00 there are 500 parcels in zone P, 300 in Q, 100 in R and 200 in S. Find the number of parcels in each zone at 12:00, assuming the same 1100 parcels stay in the warehouse. [3]
4. (d) (d) Find the exact steady-state vector. [4]

A pupil travels to school each term by Walking (W), Cycling (C) or Scooter (S). The choice in one term depends only on the choice in the previous term. After walking, the pupil never walks again: she cycles with probability 0.7 and scoots otherwise. After cycling, she walks with probability 0.4 and cycles with probability 0.35. After scooting, she walks with probability 0.25 and cycles with probability 0.5. [9]

1. (a) (a) Write down the transition matrix, with the states in the order W, C, S. [3]
2. (b) (b) Show that the chain is regular, and state the smallest power of \(\mathbf{T}\) with every entry positive. [2]
3. (c) (c) Find the eigenvalue 1 of \(\mathbf{T}\) and a corresponding eigenvector. Hence write down the steady state vector, giving its entries to 3 decimal places. [4]

A depot moves a delivery van between four sites, P, Q, R and S, every hour. The transition diagram is shown. [7]

1. (a) (a) Write down the transition matrix, with the states in the order P, Q, R, S. [3]
2. (b) (b) Find the probability that a van in S is in Q after 2 hours. [2]
3. (c) (c) Use technology to find the long-term probability that a van is in R. [2]