1 The number of cars passing a speed camera on a main road between 9.30 am and 11.30 am may be modelled by a Poisson distribution with a mean rate of 2.6 per minute.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the distribution of $X$, the number of cars passing the speed camera during a 5-minute interval between 9.30 am and 11.30 am .
\item Determine $\mathrm { P } ( X = 20 )$.
\item Determine $\mathrm { P } ( 6 \leqslant X \leqslant 18 )$.
\end{enumerate}\item Give two reasons why a Poisson distribution with mean 2.6 may not be a suitable model for the number of cars passing the speed camera during a 1 -minute interval between 8.00 am and 9.30 am on weekdays.
\item When $n$ cars pass the speed camera, the number of cars, $Y$, that exceed 60 mph may be modelled by the distribution $\mathrm { B } ( n , 0.2 )$.

Given that $n = 20$, determine $\mathrm { P } ( Y \geqslant 5 )$.
\item Stating a necessary assumption, calculate the probability that, during a given 5-minute interval between 9.30 am and 11.30 am , exactly 20 cars pass the speed camera of which at least 5 are exceeding 60 mph .
\end{enumerate}

\hfill \mbox{\textit{AQA S2 2011 Q1 [13]}}