\documentclass[a4paper,12pt]{article}
\begin{document}
\parindent=0pt
QUESTION
In a certain windy desert, sandstorms occur randomly at an
average rate of one every two days. Calculate
\begin{description}
\item[(i)]
the probability that, on a randomly chosen day, there will
be two sandstorms,
\item[(ii)]
the probability that, in a randomly chosen week, there will
be more than two sandstorms,
\item[(iii)]
the probability that, on a randomly chosen day there will
not be a sandstorm,
\item[(iv)]
the probability that, in a randomly chosen week, there will
be exactly two days on which there are no sandstorms.
\end{description}
\bigskip
ANSWER
Rate of sandstorms $\lambda =\frac{1}{2}$ Number of sandstorms per
day is
$P(\frac{1}{2})$
\begin{description}
\item[(i)]
$P(2)=e^{- \frac{1}{2}}\frac{(\frac{1}{2})^2}{2!}=0.758$
\item[(ii)]
The number of sandstorms in a week is P($\frac{7}{2}$)
\begin{eqnarray*}
\textrm{P(more than two)}&=&1-P(0)-P(1)-P(2)\\
&=&1-e^{-\frac{7}{2}}\left(1+\frac{7}{2}+\frac{(\frac{7}{2})^2}{2!}\right)\\
&=&0.679
\end{eqnarray*}
\item[(iii)]
P(0)=$e^{-\frac{1}{2}}=0.0607$
\item[(iv)]
Number of days on which there were sandstorms$\sim B(7,0.607)$
$$P(2)=\left(
\begin{array}{c}7\\2\end{array}\right)(.607)^2 (0.393)^5=.0729$$
\end{description}
\end{document}