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IB Mathematics: Applications and Interpretation HL · AHL 7: Markov Chains & Transition Matrices · AHL 7.5

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

Section A: short questions

    1. (a) \(\mathbf{T}=\begin{pmatrix}0.4 & 0.3 & 0.55 \\ 0.35 & 0.5 & 0 \\ 0.25 & 0.2 & 0.45\end{pmatrix}\)
    2. (b) A diagram with vertices R, S, T and 8 arrows: loop at R: 0.4; R to S: 0.35; R to T: 0.25; S to R: 0.3; loop at S: 0.5; S to T: 0.2; T to R: 0.55; loop at T: 0.45
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.85 & 0.4 & 0 \\ 0.15 & 0.45 & 0 \\ 0 & 0.15 & 1\end{pmatrix}\)
    2. (b) X: its column is \((0, 0, 1)\), a 1 on the diagonal and 0 elsewhere
    3. (c) 0.0825
    1. (a) 0.612
    2. (b) wet 0.611, dry 0.389
    1. (a) \(0.6s_1 + 0.15s_2 + 0.2s_3 = s_1\); \(0.25s_1 + 0.7s_2 + 0.1s_3 = s_2\); \(s_1+s_2+s_3=1\) (any two of the three row equations, with the sum)
    2. (b) \(\mathbf{s}=\begin{pmatrix}\frac{10}{33} \\ \frac{4}{11} \\ \frac{1}{3}\end{pmatrix}\)
    3. (c) 727
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.75 & 0.4 \\ 0.25 & 0.6\end{pmatrix}\)
    2. (b) 0.06
    3. (c) 0.338
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.7 & 0.4 \\ 0.3 & 0.6\end{pmatrix}\)
    2. (b) 286 active, 214 inactive
    3. (c) 286
    4. (d) \(\frac{4}{7}\)

Section B: long questions

    1. (a) \(\mathbf{T}=\begin{pmatrix}0.6 & 0.25 & 0.15 \\ 0.3 & 0.4 & 0.35 \\ 0.1 & 0.35 & 0.5\end{pmatrix}\)
    2. (b) 0.325
    3. (c) \(\begin{pmatrix}0.333 \\ 0.351 \\ 0.316\end{pmatrix}\)
    4. (d) \(\begin{pmatrix}0.336 & 0.336 & 0.336 \\ 0.351 & 0.351 & 0.351 \\ 0.313 & 0.313 & 0.313\end{pmatrix}\)
    5. (e) 31.3%
    6. (f) In the long run the chain settles to the steady state, which does not depend on the starting state
    1. (a) Stations 0 and 4. Once the chain is in an absorbing state it stays there
    2. (b) \(\mathbf{T}=\begin{pmatrix}1 & 0.6 & 0 & 0 & 0 \\ 0 & 0 & 0.6 & 0 & 0 \\ 0 & 0.4 & 0 & 0.6 & 0 \\ 0 & 0 & 0.4 & 0 & 0 \\ 0 & 0 & 0 & 0.4 & 1\end{pmatrix}\)
    3. (c) 0.288
    4. (d) 0.585
    5. (e) The long-term columns are different, because where the chain ends depends on where it starts
    6. (f) 234
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.8 & 0.3 \\ 0.2 & 0.7\end{pmatrix}\)
    2. (b) \(\lambda=1\) and \(\lambda=0.5\)
    3. (c) \(\lambda=1\): \(\begin{pmatrix}3 \\ 2\end{pmatrix}\); \(\lambda=0.5\): \(\begin{pmatrix}1 \\ -1\end{pmatrix}\) (or any multiples)
    4. (d) \(\mathbf{P}=\begin{pmatrix}3 & 1 \\ 2 & -1\end{pmatrix}\), \(\mathbf{D}=\begin{pmatrix}1 & 0 \\ 0 & 0.5\end{pmatrix}\)
    5. (e) \(0.6-0.15(0.5)^n\)
    6. (f) 0.6
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.5 & 0.1 & 0.2 & 0.25 \\ 0.25 & 0.45 & 0.1 & 0 \\ 0.05 & 0.3 & 0.4 & 0.15 \\ 0.2 & 0.15 & 0.3 & 0.6\end{pmatrix}\)
    2. (b) 0.333
    3. (c) \(\begin{pmatrix}311 \\ 190 \\ 220 \\ 379\end{pmatrix}\) (nearest whole numbers)
    4. (d) \(\mathbf{s}=\begin{pmatrix}\frac{744}{2591} \\ \frac{430}{2591} \\ \frac{505}{2591} \\ \frac{912}{2591}\end{pmatrix}\)
    1. (a) \(\mathbf{T}=\begin{pmatrix}0 & 0.4 & 0.25 \\ 0.7 & 0.35 & 0.5 \\ 0.3 & 0.25 & 0.25\end{pmatrix}\)
    2. (b) \(\mathbf{T}\) has a zero entry but every entry of \(\mathbf{T}^{2}\) is positive, so it is regular. The smallest power is 2
    3. (c) eigenvector in proportion to (0.980, 1.824, 1.000); steady state (0.258, 0.480, 0.263)
    1. (a) \(\mathbf{T}=\begin{pmatrix}0.5 & 0.1 & 0 & 0.3 \\ 0.5 & 0.2 & 0 & 0 \\ 0 & 0.7 & 0.45 & 0.2 \\ 0 & 0 & 0.55 & 0.5\end{pmatrix}\)
    2. (b) 0.15
    3. (c) 0.301

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