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\begin{document} %(
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\title{Problem Set 07}
% YOUR NAME GOES HERE under Author
\author{Your Name}
\maketitle

\begin{enumerate} %(

\item In Marc's casino, I have a game called Mediocre Poker.  In this
game, I shuffle a deck and (fairly) deal you 5 cards.  I take the
highest value of your hand and I the pay out {\em one} of the
following:
{\bf Pair or less} (Nothing),
{\bf 2 Pair} \$2,
{\bf 3 of a kind} \$5,
{\bf Straight} \$25,
{\bf Flush} \$50,
{\bf Full House} \$100,
{\bf 4 of a kind} \$500,
{\bf Straight flush} \$10,000,
{\bf Royal Flush} \$100,000.
(Remember from the class handout that there are 
123,552 ways to get a 2 Pair, 
54,912 possible 3 of a kinds, 
9,180 Straights, 
5,112 Flushes, 
3,744 Full Houses, 
624 possible 4 of a kinds, 
32 Straight flushes, and 
4 Royal Flushes.)

\begin{enumerate} %( 
\item What is the expected value of playing one game of Mediocre Poker?

% Answer below this line.

\item What is the variance of playing one game of Mediocre Poker?

% Answer below this line.

\end{enumerate} %)

\item In Marc's Casino, I've decided to automate Mediocre Poker using
a coin operated machine.  My machine accepts nickels, dimes, and
quarters.
\begin{enumerate} %( 

\item Set up a recurrence relation for the number of different ways
one can pay $5n$ cents to the machine (where the order in which the
coins are inserted matters).

% Answer below this line.

\item What characteristic equation would we have to solve to obtain a
closed form solution for the above recurrence relation?

% Answer below this line.

\item If it costs \$1 to play Mediocre Poker, how many ways are
there to feed the machine?

% Answer below this line.

\item How many ways would there be to insert 0, 1, 2, 3, 4, 5, 6, 7,
8, 9, and 10 cents, respectively, if my Mediocre Poker machine took
only 3-cent-pieces and pennies?

% Answer below this line.

\end{enumerate} %)


\item 
\begin{enumerate} %( 
\item List all 16 of the relations on $\lx\{0,1\rx\}$:

% Answer below this line.

\item BONUS: {\em Name} all of the 16 relations on $\lx\{True, False\rx\}$
(e.g., ``and'', ``only if'', ``nand''):

% Answer below this line.

\end{enumerate} %)

\end{enumerate} %)
\end{document} %)