%source: fall99 final exam problem 8
%topic: Markov Theorem

\begin{problem}[15]
A herd of cows is stricken by an outbreak of \emph{cold cow disease}.  The
disease lowers the normal body temperature of a cow, and a cow will die if
its temperature goes below 90 degrees F.  The disease epidemic is so
intense that it lowered the average temperature of the herd to $85$
degrees.  Body temperatures as low as $70$ degrees, \textbf{but no lower},
were actually found in the herd.

\bparts

\ppart[10] Based solely on the information above, state a \textbf{largest
possible} lower bound on the probability that a randomly chosen cow from
herd will have a temperature too low to survive.

\solution[\vspace{4cm}]{
Let $T$ be the temperature of a random cow.  Apply Markov's Bound to
$T-70$:
\[
\prob{T \geq 90} = \prob{T-70 \geq 20} \leq \expect{T-70}/20 = (85-70)/20 = 3/4,
\]
so
\[
\pr{T < 90} \geq \frac{1}{4}.
\]
}

%\vspace{3in}

\ppart[5] Suppose there are 400 cows in the herd.  Give an example set of
temperatures for the cows so that the probability that a randomly chosen
cow will have a fatal temperature is as small as possible.

\solution[\vspace{4cm}]{
Let 100 cows have 70 degrees and 300 have 90
degrees.  The mean is
\[
(1/4)70 + (3/4)90 = 85,
\]
as required.
}

%\vspace{2in}

\eparts

\end{problem}
