% 35 copies

\documentstyle[12pt,twoside]{article}
\input{macros-course}
\begin{document}
\handout{hw3}{Optional Homework 3}{August 2, 2001}

\begin{problems}

\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 predicted the outcome of the die
roll.  (If more than one player was correct, the money is divided
equally among the winners.  If no one was 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 players's number does not come up at all, she loses a
dollar.

\end{itemize}

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

% This problem is misstated. ARGH!!!  It should say, ``you pay the player
% $3^k$ dollars, where $k$ is the number of tails that he flipped.

\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 for each 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.

\newpage

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}
