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IB Mathematics: Applications and Interpretation HL · AHL 7: Markov Chains & Transition Matrices · AHL 7.3

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Steady state and long-term probabilities Problem set B

Use the calculator for powers of a matrix and for solving equations. Give exact answers as fractions when the question says exact, and other answers to 3 significant figures.

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. \(\mathbf{T} = \begin{pmatrix} 0.58 & 0.17 & 0.08 \\ 0.29 & 0.68 & 0.12 \\ 0.13 & 0.15 & 0.8 \end{pmatrix}\) and \(\mathbf{s}_0 = \begin{pmatrix} 0.2 \\ 0.5 \\ 0.3 \end{pmatrix}\). Find \(\mathbf{T}^{50} \mathbf{s}_0\) and \(\mathbf{T}^{100} \mathbf{s}_0\). Use them to write down the long-term state vector.

  2. \(\mathbf{T} = \begin{pmatrix} 0.43 & 0.24 & 0.33 \\ 0.37 & 0.46 & 0.09 \\ 0.2 & 0.3 & 0.58 \end{pmatrix}\). The steady-state vector is \(\mathbf{s} = (s_1, s_2, s_3)\). Use \(\mathbf{T}\mathbf{s} = \mathbf{s}\) to write down a system of three linear equations that can be solved for \(s_1, s_2, s_3\). Do not solve it.

  3. Find the exact steady-state vector by setting up and solving a system of equations.

    1. (a) \(\mathbf{T} = \begin{pmatrix} 0.73 & 0.38 \\ 0.27 & 0.62 \end{pmatrix}\)
    2. (b) \(\mathbf{T} = \begin{pmatrix} 0.35 & 0.22 & 0.12 \\ 0.28 & 0.58 & 0.32 \\ 0.37 & 0.2 & 0.56 \end{pmatrix}\)

  4. Find the exact steady-state vector for \(\mathbf{T} = \begin{pmatrix} 0.2 & 0.38 & 0.22 & 0.14 \\ 0.3 & 0.02 & 0.2 & 0.22 \\ 0.1 & 0.48 & 0.48 & 0.44 \\ 0.4 & 0.12 & 0.1 & 0.2 \end{pmatrix}\).

  5. Use your calculator to find the steady-state vector. Give the entries to 3 significant figures.

    1. (a) \(\mathbf{T} = \begin{pmatrix} 0.83 & 0.06 & 0.11 \\ 0.04 & 0.79 & 0.18 \\ 0.13 & 0.15 & 0.71 \end{pmatrix}\)
    2. (b) \(\mathbf{T} = \begin{pmatrix} 0.48 & 0.26 & 0.34 & 0.16 \\ 0.42 & 0.24 & 0.08 & 0.3 \\ 0.04 & 0.1 & 0.38 & 0.12 \\ 0.06 & 0.4 & 0.2 & 0.42 \end{pmatrix}\)

  6. \(\mathbf{T} = \begin{pmatrix} 0.4 & 0.02 & 0.32 & 0.12 \\ 0.26 & 0.36 & 0.12 & 0.4 \\ 0.32 & 0.24 & 0.08 & 0.28 \\ 0.02 & 0.38 & 0.48 & 0.2 \end{pmatrix}\) and \(\mathbf{s}_0 = \begin{pmatrix} 0.5 \\ 0.1 \\ 0.1 \\ 0.3 \end{pmatrix}\). Find \(\mathbf{s}_4\).

  7. The transition matrix \(\mathbf{T} = \begin{pmatrix} 0.64 & 0.27 \\ 0.36 & 0.73 \end{pmatrix}\) gives the probability that a commuter's next journey is by train (R) or by bus (U), with the modes in the order R, U. Write down the probability that a commuter who now takes the train takes the bus on the next journey.

  8. A forest has three kinds of tree, 1, 2 and 3. Each year the trees change with transition matrix \(\mathbf{T} = \begin{pmatrix} 0.66 & 0.12 & 0.17 \\ 0.23 & 0.79 & 0.24 \\ 0.11 & 0.09 & 0.59 \end{pmatrix}\), in that order.

    1. (a) Find the long-term probability matrix, with entries to 3 significant figures.
    2. (b) Hence state the long-term percentage of type 3.

In context

  1. A student is on time (O) or late (L) for school each day. If the student is on time today, the probability of being on time tomorrow is 0.92. If the student is late today, the probability of being on time tomorrow is 0.53.

    1. (a) Write down the transition matrix.
    2. (b) The student was late today. Find the probability that the student is on time on the fourth day after today.
    3. (c) Find the exact steady-state vector, and state the long-term probability that the student is on time.

Stretch

  1. \(\mathbf{T} = \begin{pmatrix} p & 0.28 \\ 1 − p & 0.72 \end{pmatrix}\) is a transition matrix with steady-state vector \(\mathbf{s} = \begin{pmatrix} 0.4 \\ 0.6 \end{pmatrix}\).

    1. (a) Use \(\mathbf{T}\mathbf{s} = \mathbf{s}\) to write down an equation for p.
    2. (b) Solve it to find p.

jmaths.xyz · AI HL · AHL 7.3 Steady state and long-term probabilities