\documentclass[12pt,twoside]{article}
\input{macros-course}
\begin{document}
\handout{hw4}{Optional Homework 4}{August 9, 2002}

\begin{problems}

\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}
