4.19 Transition matrices and Markov chainsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Each day a commuter either cycles (C) or takes the bus (B). If she cycles today, the probability that she cycles tomorrow is $0.7$. If she takes the bus today, the probability that she cycles tomorrow is $0.4$. On Monday she cycles. Order the states C, B.

   (a)

   The transition matrix $T$ is defined so that $\mathbf{s}_{n+1}=T\mathbf{s}_n$, where $\mathbf{s}_n$ is the column state matrix on day $n$. Which matrix is $T$?

   [1 mark]

   • A$\begin{pmatrix} 0.7 & 0.3 \\ 0.4 & 0.6 \end{pmatrix}$
   • B$\begin{pmatrix} 0.7 & 0.4 \\ 0.3 & 0.6 \end{pmatrix}$
   • C$\begin{pmatrix} 0.7 & 0.6 \\ 0.3 & 0.4 \end{pmatrix}$
   • D$\begin{pmatrix} 0.3 & 0.4 \\ 0.7 & 0.6 \end{pmatrix}$

   (b)

   Find the probability that she cycles on Wednesday (two days after Monday).

   [1 mark]

   • A$0.49$
   • B$0.70$
   • C$0.61$
   • D$0.39$

   (c)

   Find the long-term probability that she cycles on a given day.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A machine is either working (W) or faulty (F) at the start of each day. A working machine is faulty the next day with probability $0.1$. A faulty machine is working the next day with probability $0.6$. The machine is working on day 0. Order the states W, F.

   (a)

   Find the probability that the machine is faulty on day 2.

   [1 mark]

   • A$0.09$
   • B$0.04$
   • C$0.10$
   • D$0.13$

   (b)

   Find the long-term proportion of days on which the machine is faulty.

   [1 mark]

   • A$\frac17$
   • B$\frac16$
   • C$\frac67$
   • D$\frac1{10}$

   (c)

   Use your GDC to find the probability that the machine is faulty on day 3.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A website has $1000$ subscribers, each on one plan at the start of each month: Free (F), Standard (S) or Premium (P). Each month: a Free subscriber stays Free with probability $0.8$ and otherwise moves to Standard; a Standard subscriber moves to Free with probability $0.1$, stays with probability $0.7$ and otherwise moves to Premium; a Premium subscriber stays with probability $0.7$ and otherwise moves to Standard. Initially $600$ are Free, $300$ are Standard and $100$ are Premium. Order the states F, S, P.

   (a)

   Write down the transition matrix $T$, and find the number of Premium subscribers after one month.

   [3 marks]

   (b)

   Find the long-term proportion of subscribers on each plan, giving exact values.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A food truck is at either the Harbour (H) or the Market (M) each day. After a day at the Harbour, the truck is at the Harbour the next day with probability $p$. After a day at the Market, it is at the Harbour the next day with probability $0.4$. On day 1 the truck is at the Harbour. Order the states H, M.

   (a)

   (i) Write down the transition matrix $T$ in terms of $p$.
   (ii) The long-term probability that the truck is at the Harbour is $0.5$. Find the value of $p$.

   [6 marks]

   (b)

   Given that $p=0.6$:
   (i) find the eigenvalues of $T$;
   (ii) show that the eigenvector for the eigenvalue $1$ gives the long-term probabilities, and state them;
   (iii) find the probability that the truck is at the Harbour on day 4.

   [6 marks]

   Total for question 4: 12 marks

End of questions