DP Mathematics: Applications and Interpretation Questionbank

Find the transition matrix for the maze.

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.

Determine the transition matrix for this graph.

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 $\text{F}$.

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

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

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

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

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

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

Complete the following transition diagram to represent this information.

Katie works for $180$ days in a year.

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

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 $c, d\in {\mathrm{\mathbb{Z}}}^{+}$.

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

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

Find the eigenvalues and corresponding eigenvectors of T.

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 $n\in \mathbb{N}$.

What does $b$ represent in this context?

Write down the transition matrix for this Markov chain.

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.

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

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

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

Explain which part of the transition state diagram confirms this.

Write down the value of $b$.

when $n=5$.

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

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.

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

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

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

Find the steady state probability vector for this Markov chain.

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

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.

Draw a transition state diagram for this Markov chain problem.

in the long term.