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\pset{9}{November 4, 1997}{November 18, 1997}

{\bf Reading:} 4.4-4.5 of Rosen

\bigskip

\begin{problems}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 1
\problem{\bf [5 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. 

\bparts
\ppart
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?

\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?
\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 2
\problem{\bf [10 points]}

Consider a three by three array of squares as in the game of tic-tac-to.

\bparts
\ppart

If three squares are picked at random (independently and with the same
probability), what is the probability that these squares will form a line?
(horizontal, vertical, or diagonal)

\ppart
If you pick four squares at random, what is the probability that three
of them will be on a line?

\eparts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 3
\problem{\bf [10 points]}\\

\bparts

\ppart
You roll two dice.  What is the probability that their sum is not 6?

\ppart 
You roll three dice. What is the probability that that any of the 
{\em pairwise} sums is not 6?

\eparts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 4
\problem {\bf [10 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 
$Pr(A|B\cap C)$ is large? Give a counterexample or a proof to justify 
your answer.
\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 5
%The Prisoner's Dilemma.
\problem{\bf [5 points]}

Three Prisoners $A$,$B$, and $C$ have applied for parole.  The parole
board has decided to randomly choose two of the three to be released,
 and the prisoners know
this but not which two.  A warder friend of prisoner $A$ knows who are
to be released.  $A$ realizes that it would be unethical to ask if he
is going to be released, but thinks of asking for the name of {\em one
  prisoner other than himself} who is to be released. Before he asks
his chances are $2/3$, but he thinks that if he asks and the warder
says ``$B$ will be released'', his own chances will become $1/2$ since
either $A$ or $C$ will be the only remaining candidates.  So $A$
decides not to reduce his chances by asking.  Is $A$'s argument correct?
Show what his chances are after asking.
You may assume that if the warder has a choice of which name to reveal,
then he picks one at random.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 6
\problem {\bf [10 points]}

We know from statistics that when a policeman gets shot he/she survives with
probability $0.8$ if he/she wears a bullet-proof vest and survives with
probability $0.12$ if the policeman doesn't wear a vest.
Given that 15\% of the policemen wear bullet-proof vests, what is the
probability that a random policeman was wearing a bullet-proof vest if he/she
survived a firearm attack?
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 7
\problem {\bf [10 points]}

$A$, $B$, and $C$ are to fight a three-cornered pistol duel.  They all know 
that $A$'s chance of hitting the target is $0.3$, $C$'s is $0.5$ and $B$ 
never misses.  They are to shoot in the sequence $A,B,C$ and fire at their 
choice of target until one person is left standing.  Show that $A$ is better
off missing deliberately the first shot than attempting to hit either $B$ or 
$C$.  Assume that every player plays optimally.

({\em Hint: consider the situation where $B$ is dead and $C$ shoots next.
What is the probability $P$ that $A$ ultimately survives? 
Show that $P = .5(.3+.7P)$.})

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 8
%Independence
\problem{\bf [10 points]}\\

\bparts

\ppart
Show that if two events $A$ and $B$ are independent, then the events $A$ 
and $\overline B$ are also independent.

\ppart
Two numbers $x$ and $y$ are selected at random from the set
$\{1,\ldots,100\}$.  Let the events $A$, $B$ and $C$ be defined
as follows: $A = \{x > 50\}$, $B = \{y > 50\}$ and $C=\{x > y \}$.
Are the events $A$ and $B$ independent? Are $A$ and $C$ independent?

\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 9
\problem{\bf [10 points]}

You are given a coin of bias $p$ (prob of heads = $p$; tails = 1-$p$,
$0 < p < 1$).  The value of $p$ is not known and you are not allowed to guess
by sampling.

Using this, how can you get a coin of bias:

\begin{enumerate}

\item $p^2$

\item ${1 \over 2} p^2 + {1 \over 4} p$

\item ${1\over 2} - {1 \over 4}p$


\end{enumerate} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 10
\problem{\bf [10 points]}

Consider three loaded dice such that

Die A always comes up 3\\
Die B is 2 with prob $p$ and 5 otherwise\\
Die C is 4 with prob $p$ and 1 otherwise.\\

Find a value for $p$ such that A beats B, B beats A, and C beats A all with
probability $p$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 11
\problem{\bf [10 points]}

In a complex system with 100,000 different components.
The system contains 10 critical components, each failing with probability
$1/1000$ and 100 secondary components each failing with probability $1/100$.
It is known that secondary failures are pairwise independent.
The system crashes if one critical component or two secondary components
fail. 

\bparts

\ppart
Determine an upper
bound on the probability that the system will crash.

\ppart 
Determine a lower bound on the probability that the system will crash.

\ppart
Where is the biggest weakness? In the critical, or in the secondary components?

\eparts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\problem 

%In a sequence of Bernoulli trials with success probability
%$p$, find the probability that $a$ successes will occur before $b$
%failures.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\problem 

%In this problem, we will prove that there is a tournament
%graph with the following bizarre property: for every set of $k$ people
%in the tournament, there is a certain person who has beaten all of
%them---this makes it hard it identify the best $k$ players!  We'll use
%a technique called the {\em probabilistic method} where you prove
%something exists by showing that that the probability it exists is not
%0.

%Consider an $n$-vertex tournament that is generated randomly by
%assigning each edge a random (equally probable) direction.

%\bparts

%\ppart Consider a certain set of $k$ people.  What is the probability a
%particular other person beats all of them?

%\ppart What is the probability that {\em nobody} beats all of them?

%\ppart Using the union bound, upper bound the probability that there
%is some set of $k$ people for which nobody else beats all of them?

%\ppart.   Using the identities $(1-x)< e^{-x}$ and ${n \choose k} <
%(en/k)^k$, simplify your answer to the previous problem.

%\ppart Find the smallest $n$ (as a function of $k$) that you can for
%which the above quantity is less than $1$.

%\ppart Show that there is a tournament on $n$ vertices ($n$ you just
%found) with the claimed bizarre property.

%\eparts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{problems}

\end{document}


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