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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}

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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)}&amp;=&amp;1-P(0)-P(1)-P(2)\\
     &amp;=&amp;1-e^{-\frac{7}{2}}\left(1+\frac{7}{2}+\frac{(\frac{7}{2})^2}{2!}\right)\\
     &amp;=&amp;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}

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