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\pset{10}{November 14, 1995}{November 21, 1995}

{\bf Reading:} Pages 261-275 of {\it
Discrete Mathematics and its Applications} by Kenneth H. Rosen.

\bigskip
\begin{problems}

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\problem
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 hear Andrew (Carol) placing a
``Nerd Pride'' pocket protector underneath Box~1.  What should you do
to try to win the prize (the ice cream)?  What is the probability that
you win?
\ppart Consider the game where Monty always puts the prize behind  
 the same door (say box 1) but you pick your doors uniformly at
random. Show that this version of the game is the same as the original
one where both Monty and you picked the doors at random. (By ``same
as'' we mean that the strategies ``don't-switch,'' ``switch'' have the
same probabilities of winning.)
\ppart
Suppose Monty is determined to make you lose.  In particular, if you
initially choose a losing door, he will 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

\problem
\bparts
\ppart Suppose you are playing the {\it Lets Make a Deal} game with
4 doors rather than~3.  Say Monty and you choose doors according to
the uniform distribution.  After your initial choice, Monty reveals a
loser and offers you the chance to switch.  Subsequently, he reveals
another loser and again offers you the chance to switch.  Compute the
probability of winning for each of the four possible strategies: (1)
stick both times, (2) switch both times, (3) stick the first time,
switch the second time, and (4) switch the first time, stick the
second time.  Which strategy is the best?
\ppart Consider the 3 door version of the game. Suppose you know that 
Monty always reveals the lowest numbered door available. Does this new
information help you? If not, why not? If yes, what should your
strategy be and why?
\eparts


\problem
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 the criminal
in this case has a beard (given that the criminal was drawn uniformly
at random from the entire population)?

\problem
Suppose $A,B,C$ are independent events. Show that any of the combinations
\[ A^c, B,C; \, A, B^c,C; \cdots; A^c, B^c,C; \cdots; A^c,B^c,C^c \]
are also independent. Furthermore, show that $A$ and $B \cup C$ are 
independent.


\problem
Once upon a time, a soldier was sentenced to death.  To show his kindness,
the king decided to give him another chance. He gave the soldier $50$ black
balls and $50$ white balls and $2$ pots and told him to divide the balls
between the two pots in whatever way he liked.  Then the king would come
back and randomly take away one pot and mix the balls in the other. The
soldier would then pick a ball from the remaining pot and live if and only
if it was white. What strategy should the soldier take? Prove that it is the
best.

\iffalse 
\problem
Consider a country where people favor boys over girls and each family
will keep having babies until they have a boy. (And then they stop
having kids.)  Assume that genetically boys and girls are equally
likely to be born, and that the death rate is the same for both sexes.
Will there be a bias in that country's population in the long run?
Explain your answer.
\fi

\problem
Mr. James W. Pitman received a letter: \\
`` $\cdots$ to inform you that one of the following three people has
 won a New Datsun Sentra: Collin Andrus; James W. Pitman; Larry Abbott. 
$\cdots$"\\
`` $\cdots$ you will receive one of the following for sure:\\
$(1)$ hairdryer; $(2)$ opportunity to be a partner in Bonanza Gold Mine Inc; $(3)$ Datsun Nissan Sentra.\\
To claim your gift, call $\cdots$"\\
The fine print at the back of the letter says ``your chances of
winning a hairdryer or a Datsun Nissan Sentra are $1$ in $10,000$ and
your chances of winning an opportunity to be a partner in Bonanza Gold
Mine Inc. are $9,998$ in $10,000$.''

Let us assume that this is an honestly conducted sweepstakes, with
$10,000$ participants and exactly one Datsun being awarded. Now that
Mr. Pitman knows that the winner is one of three individuals (and he
is on the list), the rules of conditional probability imply that he
has a $\frac{1}{3}$ chance of winning the Datsun. True or False?
Explain!

\end{problems}

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