5 A local hockey league has three divisions. Each team in the league plays in a division for a year. In the following year a team might play in the same division again, or it might move up or down one division.

This question is about the progress of one particular team in the league. In 2007 this team will be playing in either Division 1 or Division 2. Because of its present position, the probability that it will be playing in Division 1 is 0.6 , and the probability that it will be playing in Division 2 is 0.4 .

The following transition probabilities apply to this team from 2007 onwards.

\begin{itemize}
  \item When the team is playing in Division 1, the probability that it will play in Division 2 in the following year is 0.2 .
  \item When the team is playing in Division 2, the probability that it will play in Division 1 in the following year is 0.1 , and the probability that it will play in Division 3 in the following year is 0.3 .
  \item When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 .
\end{itemize}

This process is modelled as a Markov chain with three states corresponding to the three divisions.\\
(i) Write down the transition matrix.\\
(ii) Determine in which division the team is most likely to be playing in 2014.\\
(iii) Find the equilibrium probabilities for each division for this team.

In 2015 the rules of the league are changed. A team playing in Division 3 might now be dropped from the league in the following year. Once dropped, a team does not play in the league again.\\
-The transition probabilities from Divisions 1 and 2 remain the same as before.

\begin{itemize}
  \item When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 , and the probability that it will be dropped from the league is 0.1 .
\end{itemize}

The team plays in Division 2 in 2015.\\
The new situation is modelled as a Markov chain with four states: 'Division1', 'Division 2', 'Division 3' and 'Out of league'.\\
(iv) Write down the transition matrix which applies from 2015.\\
(v) Find the probability that the team is still playing in the league in 2020.\\
(vi) Find the first year for which the probability that the team is out of the league is greater than 0.5 .

\section*{ADVANCED GCE UNIT MATHEMATICS (MEI)}
4757/01

\section*{Further Applications of Advanced Mathematics (FP3) }
\section*{THURSDAY 14 JUNE 2007}
Afternoon\\
Time: 1 hour 30 minutes\\
Additional materials:\\
Answer booklet (8 pages)\\
Graph paper\\
MEI Examination Formulae and Tables (MF2)

\begin{itemize}
  \item Write your name, centre number and candidate number in the spaces provided on the answer booklet.
  \item Answer any three questions.
  \item You are permitted to use a graphical calculator in this paper.
  \item Final answers should be given to a degree of accuracy appropriate to the context.
\end{itemize}

\begin{itemize}
  \item The number of marks is given in brackets [ ] at the end of each question or part question.
  \item The total number of marks for this paper is 72.
\end{itemize}

\begin{itemize}
  \item Read each question carefully and make sure you know what you have to do before starting your answer.
  \item You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
\end{itemize}

\hfill \mbox{\textit{OCR MEI FP3  Q5}}