Using your answer to (a), or otherwise, find the long-term probability of the switch being in state $\text{A}$. Give your answer in the form $\frac{c}{d}$, where $c, d\in {\mathrm{\mathbb{Z}}}^{+}$.

Explain why for any transition state diagram the sum of the out degrees of the directed edges from a vertex (state) must add up to +1.

Find the eigenvalues and corresponding eigenvectors of T.

Explain which part of the transition state diagram confirms this.

What does $b$ represent in this context?

when $n=5$.

in the long term.

Write down the value of $b$.

Draw a transition state diagram for this Markov chain problem.

Find the steady state probability vector for this Markov chain.

Hence write down the number of customers that company X can expect to have in the long term.

We know that she went out for lunch on a particular Sunday, find the probability that she went out for lunch on the following Tuesday.

It is sunny today. Find the probability that it will be sunny in three days’ time.

Find the eigenvalues and eigenvectors of $\mathit{T}$.

Write down the matrix $\mathit{D}$.

Explain why having a steady state probability vector means that the matrix M must have an eigenvalue of $\lambda =1$.

Show that $\lambda =1$ is always an eigenvalue for M and find the other eigenvalue in terms of $a$ and $b$.

Find ${\mathbf{v}}_1\text{,}{\mathbf{v}}_2\text{,}{\mathbf{v}}_3\text{,}{\mathbf{v}}_4$.

Explain how your answer to part (f) fits with your answer to part (c).

Write down the transition matrix for this Markov chain.

Hence write down matrices P and D such that T = PDP⁻¹.

Find an expression for the number of customers company X has after $n$ years, where $n\in \mathbb{N}$.

Find the steady state probability vector for M in terms of $a$ and $b$.

Find the transition matrix for the maze.

Find the steady state probability vector for this Markov chain problem.

Find the minimum number of tosses of the coin that Abi will have to make to be at least 95% certain of having finished the game by reaching state C.

Write down a transition matrix T representing the movements between the two companies in a particular year.

Hence find the long-term percentage of sunny days in Vokram.

A scientist sets up the robot and then leaves it moving around the maze for a long period of time.

Find the probability that the robot is in room $\text{B}$ when the scientist returns.

Write down the transition matrix M, for this Markov chain problem.

Hence, deduce the form of ${\mathbf{v}}_n$.

Determine the transition matrix for this graph.

Comment on your answer to part (b), referring to at least one limitation of the model.

If the mouse was left to wander indefinitely, use your graphic display calculator to estimate the percentage of time that the mouse would spend at point $\text{F}$.

Write down the matrix $\mathit{P}$.

Katie works for $180$ days in a year.

Find the probability that Katie cycles to work on her final working day of the year.