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\centerline{Mathematics 375 -- Probability and Statistics 1}
\centerline{Problem Set 3}
\centerline{{\bf due:} In class, Friday September 23}
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A)  Much of the mathematical theory of probability was first developed 
(ironically enough)
to analyze what happens in {\it games of chance} including various
card and dice games used for gambling.  One famous example is the 
dice game ``craps'' (a mainstay of mobster movies, {\it The Sopranos}, etc.)
In case you have never played, the rules of craps are the following:
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\item{1)}  At the start of the game, the player rolls two dice.
\item{2)}  If the first total is 7 or 11, then the player wins immediately.
\item{3)}  If the first total is 2,3, or 12, then the player loses
immediately.
\item{4)}  If the first total is anything else, that becomes the 
player's ``point''.  The player
rolls the dice again (repeatedly) until either the point total
is obtained again, in which case the player wins, or else a 7 is rolled, in 
which case the player loses.  {\it Any number of additional rolls
is possible in this case; the game continues until either the point
is rolled again, or a 7 is rolled}.
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The Maple package for our class now contains a 
craps simulator procedure that can be used to generate any number
of craps games and study the outcomes.  {\it Note:  I just added
this yesterday (9/18/03), so you will need to download the package
again to get a version including this new procedure.}  After you have
loaded the package, you call the procedure with a command of the form
${\it Craps(n,verbose);}$ where $n$ is the number of games you 
want to simulate, and $verbose$ (true or false) tells Maple 
how much output to print out as the games are played.
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\item{$\bullet$} {\it Craps(10,true);} -- plays 10 full games and 
prints all rolls as it goes along.  The final output is 
a list of 10 0's and 1's (0 = player loses in that game; 1 = player wins).
\item{$\bullet$} {\it Craps(100,false);} -- plays 100 full games
and generates the final output, but doesn't show all the individual rolls.
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\item{1)}  Use the {\it Craps} procedure to play 5 games, print out
all the rolls, and explain the outcome of each of the games.
\item{2)}  Experiment with larger numbers of games (you'll probably
want to do lots!).  What can you 
say about the apparent {\it probability} that the player wins an
individual game?
\item{3)}  Now, we'll analyze the game using the probability concepts
we have learned.  We'll assume the dice used are ``fair'' (i.e. the
probability of each possible number on each die is 1/6).  
\itemitem{a)}  What is the probability of winning immediately (on the
first roll)?  What is the probability of losing immediately?
\itemitem{b)}  Now the tricky part.  What is the probability of
winning if your point was a 4 or a 10?  a 5 or a 9?  a 6 or an 8?
(Recall, the game can go on for any number of rolls!)
\itemitem{c)}  What is the total probability of winning?
How well does this match your simulation results?
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\noindent
Note:  Casino games are not likely to be popular if the player
wins too rarely.  Your answer for part 3 should give some evidence why 
``craps'' is a popular game, but one that casino operators can
make money on too(!)
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\noindent
{\it From the text:}  2.61, 63, 64, 66, 68, 69, 73, 80, 84, 87, 93, 99, 102, 107, 3.1,3
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