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\begin{problems}

%BEGIN Kavita's problems
%END   Kavita's problems

%BEGIN Joe's problems
%END   Joe's problems

%BEGIN Ray's problems

%%%%%%%%%%%%%%%% S99 ps10


\problem{\bf (12 points)}

Lockheed-Martin is currently in the process of designing the new space
shuttle, the ``Venture Star''.  Suppose that it has $100,000$ parts
and the probability that each part will fail is $1/1000$.  Redundancy
has been designed into the spacecraft so that it will fly as long as
no more than $125$ parts fail.  The failure of each part is
independent of the failure of any other part.  Prove that the
probability that Venture Star will fly successfully is at least
$99\%$.

\solution{You have to upper bound the probability that less than
$100,000-125=99,875$ parts will be OK.  This upper bound turns out to be
slightly less than $1 \%$.  You take $n=100,000$,
$\alpha=99,875/100,000=0.99875$ and $p=1-1/1000=0.999$ (then $\alpha<p$),
and compute an upper bound for $F_{n,p}(\alpha n)$ using Theorem 3.1 from
Lecture 21 in the lecture notes.  First you should get that
$f_{n,p}(\alpha n)\approx{0.001972}$.  Then
\[F_{n,p}(\alpha n)
\leq \frac{1-\alpha}{1-\frac{p}{\alpha}}f_{n,p}(\alpha n) \approx
4.995 * 0.001972 \approx 0.0098.\]
}



%END   Ray's problems

%BEGIN Ivan's problems (spring '98)
%END   Ivan's problems

%BEGIN George's problems
%END   George's problems

%BEGIN Josh's problems
%END   Josh's problems

%BEGIN Tina's problems
%END   Tina's problems


\end{problems}

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