The number of telephone calls received, during an $8$-hour period, by an IT company that request an urgent visit by an engineer may be modelled by a Poisson distribution with a mean of $7$.

\begin{enumerate}[label=(\alph*)]
\item Determine the probability that, during a given $8$-hour period, the number of telephone calls received that request an urgent visit by an engineer is:
\begin{enumerate}[label=(\roman*)]
\item at most $5$; [1 mark]
\item exactly $7$; [2 marks]
\item at least $5$ but fewer than $10$. [3 marks]
\end{enumerate}
\item Write down the distribution for the number of telephone calls received each hour that request an urgent visit by an engineer. [1 mark]
\item The IT company has $4$ engineers available for urgent visits and it may be assumed that each of these engineers takes exactly $1$ hour for each such visit.

At $10$am on a particular day, all $4$ engineers are available for urgent visits.
\begin{enumerate}[label=(\roman*)]
\item State the maximum possible number of telephone calls received between $10$am and $11$am that request an urgent visit and for which an engineer is immediately available. [1 mark]
\item Calculate the probability that at $11$am an engineer will not be immediately available to make an urgent visit. [4 marks]
\end{enumerate}
\item Give a reason why a Poisson distribution may not be a suitable model for the number of telephone calls per hour received by the IT company that request an urgent visit by an engineer. [1 mark]
\end{enumerate}

\hfill \mbox{\textit{AQA S2 2010 Q5 [13]}}