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\pset{10}{November 14, 1996}{Start of Class, Tuesday, November 19, 1996}

{\bf Reading:} Lecture Notes on {\em Probability}.

\bigskip

\begin{problems}

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\problem {\bf [20 points]}

Consider playing the {\it Lets Make a Deal}, where there are three prizes:
a ``Gift Certificate'' from Toscanini, a ``Nerd Pocket Protector'', and a 
``Nerd Pride Button''. Your goal is to win the gift certificate. 

Suppose you (the contestant) happen
to know that Carol always reveals the ``Nerd Pocket Protector'', if you 
did not choose the box covering it. \\
Does this additional information change the odds of winning for you? 
Is there a better strategy? By how much (if any) does this information
increase your chances of winning?

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\problem {\bf [20 points]}

Consider playing the {\it Lets Make a Deal} game with the following
additional information.
\bparts
\ppart
You are playing according to the original assumptions, but before
you make your initial choice, you happen to see Carol placing a 
``Nerd Pride Button'' underneath Box~1.  What should you do to try to win 
the prize (the ice cream)?  What is the probability that you win?
\ppart
Suppose Monty is determined to make you lose. 
In particular, if you initially choose a losing door, 
he may decide to end the game right there, opening your door to reveal
the goat you have ``won'', never giving you the chance to switch doors.
What is the probability of winning if you play the ``don't-switch strategy''?  
What if you play the ``switch strategy''?
\eparts

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\problem {\bf [20 points]}

In a complex system with 1,000 different components, each component is
known to fail with probability at most $1/10,000$. Determine an upper
bound on the probability that one or more components in the system fail.

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\newpage

\problem {\bf [20 points]}

A witness to a crime says that the criminal had a beard.
We know from experience that if a criminal has a beard, 
witnesses will say so with probability~$0.8$, 
and if the criminal does not have a beard, witnesses will say that
he does with probability $0.12$.
If 15\% of the population has beards, what is the probability 
that a random criminal has a beard if the witness says he does?

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%%%	From Drake's book.
%\problem {\bf [20 points]}	

%To the best of our knowledge, with probability 0.8 Al is guilty of 
%the crime for which he is about to be tried.
%Bo and Cy, each of whom knows whether or not Al is guilty,
%have been called to testify.
%Bo is a friend of Al's and will tell the truth if Al is innocent
%but will lie with probability 0.2 if Al is guilty.
%Cy hates everybody but the judge and will tell the truth if Al is guilty
%but will lie with probability 0.3 if Al is innocent.
%\begin{problemparts}
%\problempart
%Determine the probability that the witnesses give conflicting testimony.
%\problempart
%Which witness is more likely to commit perjury?
%\end{problemparts}

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\problem {\bf [20 points]}	

\bparts
\ppart
Prove that $Pr(\overline{A}|C) = 1 - Pr(A|C)$ for any events $A$ and $C$.
\ppart
Is it possible for $Pr(A|B)$ and $Pr(A|C)$ to be very small, if 
$Pr(A|B\wedge C)$ is large? Give a counterexample or a proof to justify 
your answer.
\eparts

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\end{problems}

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