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\centerline{MONT 106N -- Identifying Patterns}
\centerline{Group Discussion -- Chance}
\centerline{November 18, 2009}
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\noindent
{\it Background}
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Today, we want to think about and experiment with a situation involving
chances, modeled on a once-popular old-time TV show called
{\it Let's Make a Deal}.  Here's the situation:
\begin{itemize}
\item The game show host, a slightly sleazy huckster type
named {\it Monty Hall}, shows the contestant {\it three closed
doors}.  He says that a valuable prize (typically a car) is
behind one of them, while the other two doors conceal worthless
joke ``prizes'' (typically live goats(!)).  
\item The contestant chooses one of the doors, but the door
{\it is not opened} to reveal the prize behind it at that point.
\item Monty, {\it who knows where the valuable prize is}, opens
one of the doors concealing one of the {\it worthless} prizes.
\item At this point, Monty gives the contestant the chance 
to either {\it stay with the door they chose originally}, or 
else {\it switch to the other unopened door}.
\item After the contestant makes his or her choice, that door
is opened and the prize is revealed.
\end{itemize}
The question is:  Is it a better strategy for the contestant
to {\it stay with their original choice of door}?  Or is it
a better strategy for them to {\it switch to the other 
unopened door}?  Or perhaps might it be best to mix up the
two possibilities, maybe by alternating between staying and
switching?
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\noindent
{\it Discussion Questions}
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\begin{enumerate}
\item[A.]   Do you have an {\it intuitive feeling} what the best
strategy should be for the contestant?  If so, what do you think?  Or should it
matter which way the contestant makes his or her choice?  
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\item[]  To get some data to analyze, we are going to {\it simulate}
the {\it Let's Make a Deal} game with slips of paper.  In your group, designate
one person as ``Monty'' for each of the following parts.  Your group
will receive three slips of paper, one marked ``car'' on one side,
and two marked ``goat'' on one side.  ``Monty'' will then place the three slips
of paper
in some random order (that he or she {\it remembers}!) in front of a second team
member who will be the contestant.  The contestant will select one slip,
{\it without turning it over}.  At least one of the unchosen slips
is a ``goat'' at this point.   ``Monty'' will then turn over one of the 
 ``goats'' and the contestant will choose to hold onto
the first slip or to switch, as below.  Then that slip will be turned
over and the third person will record whether the game was a win for
the contestant (he or she got the ``car''), or a loss (he or she got the 
other ``goat'').   Rotate the roles of ``Monty,'' the contestant, and the 
record keeper to keep everyone interested!
\item[B.]   Play 25 rounds of the game, having the contestant {\it always
hold onto their original card} when offered the opportunity to switch, 
and count the number of ``wins.''
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\item[C.]  Play 25 rounds of the game, having the contestant {\it always
switch to the other unknown card} when offered the opportunity to switch, 
and count the number of ``wins.''
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\item[D.]  Play 25 rounds of the game, having the contestant {\it alternate
between holding onto their original card and switching} when offered the opportunity to switch, 
and count the number of ``wins.''
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\item[E.]  Which strategy gave the most wins for the contestant?  Was this
a surprise?
\end{enumerate}
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\noindent
{\it Assignment}
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Group writeups due at the end of class.
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