Sign in to jmaths to print this page or save it as a PDF: jmaths-8ce638778fd6.herokuapp.com/maths/ai-hl/markov-chains-transition-matrices/ahl-7-1/problems/b/

jmaths

IB Mathematics: Applications and Interpretation HL · AHL 7: Markov Chains & Transition Matrices · AHL 7.1

Name Date

Transition matrices and diagrams Problem set B

Write each matrix with columns as the start state and rows as the end state, and check that every column adds to 1. No calculator is needed.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A Markov chain has two states, P and Q. Write down the transition matrix in each case.

    1. (a) \(P(P \to P) = 0.58\); \(P(Q \to P) = 0.09\).
    2. (b) \(P(P \to Q) = 0.76\); \(P(Q \to Q) = 0.33\).

  2. A Markov chain has three states, A, B and C. The known probabilities are: \(P(A \to A) = 0.38\); \(P(A \to B) = 0.47\); \(P(B \to A) = 0.11\); \(P(B \to B) = 0.64\); \(P(C \to B) = 0.21\); \(P(C \to C) = 0.5\). Write down the transition matrix.

  3. A Markov chain has four states, A, B, C and D. The probabilities of changing state are: \(P(A \to B) = 0.14\); \(P(A \to C) = 0.26\); \(P(A \to D) = 0.19\); \(P(B \to A) = 0.33\); \(P(B \to C) = 0.09\); \(P(B \to D) = 0.3\); \(P(C \to A) = 0.06\); \(P(C \to B) = 0.43\); \(P(C \to D) = 0.34\); \(P(D \to A) = 0.13\); \(P(D \to B) = 0.21\); \(P(D \to C) = 0.52\). Find the transition matrix.

  4. Draw the transition diagram for each matrix. The rows and columns are in the order of the states shown.

    1. (a) States P, Q: \(\mathbf{T} = \begin{pmatrix} 0.81 & 0.46 \\ 0.19 & 0.54 \end{pmatrix}\)
    2. (b) States A, B, C: \(\mathbf{T} = \begin{pmatrix} 0.58 & 0.27 & 0.12 \\ 0.42 & 0.23 & 0 \\ 0 & 0.5 & 0.88 \end{pmatrix}\)

  5. Write down the transition matrix for the transition diagram, with the states in the order P, Q.

    0.410.590.180.82PQ

  6. The transition diagram has three states. Find its transition matrix, with the states in the order A, B, C, and name the absorbing state.

    10.30.50.20.10.450.45ABC

  7. Write down the transition matrix for the transition diagram, with the states in the order W, X, Y, Z.

    0.40.60.50.20.30.150.350.50.350.250.4WXYZ

In context

  1. Three radio stations, P, Q and R, compete for listeners. Each month, 65% of the listeners of P stay with P and 25% move to Q. Of the listeners of Q, 55% stay with Q and 30% move to R. Of the listeners of R, 20% move to P and 50% move to Q. Write down the transition matrix for the monthly movement of listeners, with the stations in the order P, Q, R.

Stretch

  1. Answer each part.

    1. (a) Explain why \(\begin{pmatrix} 0.63 & 0.34 \\ 0.48 & 0.66 \end{pmatrix}\) cannot be a transition matrix.
    2. (b) \(\mathbf{T} = \begin{pmatrix} 0.22 & 0.36 & a \\ 0.48 & b & 0.14 \\ c & 0.27 & 0.42 \end{pmatrix}\) is a transition matrix. Find \(a\), \(b\) and \(c\).

  2. A plant is checked each week. It is healthy (H), infected (I) or dead (D). A healthy plant is still healthy the next week with probability 0.88, and infected with probability 0.12. An infected plant recovers to healthy with probability 0.17 and dies with probability 0.34. A dead plant stays dead.

    1. (a) Write down the transition matrix, with the states in the order H, I, D.
    2. (b) List the loops and arrows of the transition diagram.
    3. (c) State the absorbing state and say how you can see it in the matrix.

jmaths.xyz · AI HL · AHL 7.1 States, transition matrices and transition diagrams