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\handout{hw2}{Optional Homework 2}{August 2, 2005}

\begin{problems}

\problem You have a hat in which there are three coins, identical
except for color.  One coin is gold on both sides, one coin is silver
on both sides, and one coin is gold on one side and silver on the
other.  You withdraw one coin at random, look at one side, and see
that it is gold.  What is the probability that the other side is gold?

\problem A bag contains either a black marble or a white marble, each
with $50\%$ probability.  A white marble is added to the bag, the bag
is shaken, and one of the two marbles in the bag is removed.  The
removed marble proves to be white.  What is the probability that the
marble remaining in the bag is also white?

\problem A box contains two seemingly identical coins.  One is fair
(the probability of heads is $1/2$), but the other is biased with a
$2/3$ probability of heads.  One of the coins is tossed once,
resulting in heads.  The other is tossed three times, resulting in two
heads and one tail.  Which coin is more likely to be the biased one?

\problem (\textit{Hard.})  An ordinary deck of $52$ cards is shuffled
randomly and placed face down on the table.  The cards are turned over
one at a time.  Before each card is turned over, you are given the
opportunity to say ``now.''  When you say ``now,'' if the next card
turned over is black, you win.  If it is red, or if you go through the
whole deck without saying ``now,'' you lose.  

The strategy of saying ``now'' before the very first card is turned
over obviously gives you a $50$-$50$ chance of winning.  Is there a
better strategy, one that allows you to win more than half the time?
What is the probability of winning with the best strategy?

\problem (\textit{Hard.}) Someone writes two different real numbers on
two cards and places them face down on the table.  Your goal is to
choose the card with the larger number.  Obviously, you can win at
this game half the time by randomly choosing one of the cards.  But,
suppose that you are given the chance to turn over either one of the
cards and look at the number on it before choosing which card has the
larger number.  Is there a strategy for winning strictly more than
half the time?

\problem A publisher gives the proofsheets of a new book to two
different proofreaders, who read it separately.  The first proofreader
finds $60$ misprints.  The second proofreader finds $70$ misprints of
which $50$ are the same as misprints found by the first proofreader.
Make some reasonable assumptions about the probabilistic process by
which the proofreaders find misprints in order to determine the
expected number of misprints remaining undetected in the book.

\problem Alice, Bill, and Carol like to play the following game.  Each
player puts two dollars into a hat and then secretly writes a number
between 1 and 6 on a piece of paper.  Someone then rolls an ordinary
die.  The players reveal their numbers, and the money in the hat is
given to the player who correctly predicts the outcome of the die
roll.  If more than one player is correct, the money is divided
equally among the winners.  If no one is correct, everyone takes their
money back.  

Over the course of many games, Carol notices that she is slowly, but
steadily losing money.  She makes sure that the money is distributed
according to the rules.  She tests the die to confirm that it is fair.
She ensures that her guesses are secret.  She tries many different
guessing strategies.  But she keeps losing money.  Why?

\problem In the game of Chuck-a-Luck, the player picks a number
between 1 and~6.  The game operator then rolls three dice.  The
outcome of the game is determined as follows:

\begin{itemize}

\item If the player's number comes up once, she wins one dollar.

\item If the player's number comes up twice, she wins two dollars.

\item If the player's number comes up three times, she wins $p$
dollars.

\item If the player's number does not come up at all, she loses a
dollar.

\end{itemize}

For what value of $p$ is the game fair?

\problem You are running the following game at a carnival.  A player
flips a coin until he gets heads.  Then, you pay the player $3^k$
dollars, where $k$ is the number of tails that he flipped.  How much
should you charge people to play this game if you want the game to be
fair?

\problem Suppose that we play the following game.  I put some money in
one envelope and twice as much money in a second envelope.  Now you
pick an envelope at random and count how much money is inside.
Suppose you find $x$ dollars there.  Now, you can either keep the $x$
dollars or else take the unknown amount of money in the other
envelope.

If you take the $x$ dollars, then your expected payoff is, of course,
$x$ dollars.  On the other hand, if you switch, then with probability
$1/2$ you win $x/2$ dollars and with probability
$1/2$ you win $2x$ dollars.  Your expected payoff is therefore
\begin{eqnarray*}
\frac{1}{2} \cdot \frac{x}{2} + \frac{1}{2} \cdot 2x
	& = & \frac{5}{4} x
\end{eqnarray*}
Thus, the analysis indicates that switching improves your expected
payoff by $25$ percent.  Argue that this conclusion is obviously
wrong, and then pinpoint the precise error in the analysis.

\end{problems}

\end{document}

