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\centerline{MONT105Q -- Mathematical Journeys}
\centerline{The ``Monty Hall Problem''}
\centerline{March 4, 2016}
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\noindent
{\it Background}
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On the TV game show {\it Let's Make a Deal}, the old host Monty Hall would frequently
play the following game with contestants:
\begin{itemize}
\item The contestant would be shown three large closed doors and told that behind
one was a \emph{new car}, while \emph{live goats} were behind the other two.
\item The contestant would select one of the doors.
\item At this point Monty (who always \emph{knew} where the car and the goats were)
would reveal one of the goats, behind one of the doors the contestant \emph{had not picked} -- note
that with two goats, there was always at least one such door to reveal.
\item He then offered the contestant the option of \emph{staying with the original 
door they had picked} or \emph{switching to the other unopened door}.
\item When they made their choice, what was behind the door they finally selected was
revealed and the game concluded.  
\end{itemize}
(In case you're wondering, when the contestant ``won'' a goat, 
I don't think the show ever discussed what happened next.  I 
suppose it's possible some contestants might have really wanted
a goat instead of a car; others may have been at least willing to settle for a goat.  
But in most cases, the show probably allowed the contestant 
to go home without the goat even if they had ``won'' it(!))

In 1990, in her \emph{Parade} magazine column, Marilyn Vos Savant responded to a 
question from a reader and explained that the best strategy for the contestant
was to \emph{always switch doors when offered the choice}.  Immediately hundreds
of people, including a bunch of well-known professional mathematicians wrote to 
her saying she had made a mistake.  They claimed there shouldn't be any difference between
switching or keeping the original choice because at that point the chances
were 50/50 the person had the door with the car.

To start today, we want to do a simulation to gather some evidence about who
may have been right about this.
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\noindent
{\it Directions}
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\begin{itemize}
\item Work in groups of two or three (5 groups of three, assuming everyone is present).
\item In each group, one person will be Monty Hall, the other will be a contestant, and
the third will be a recorder of the results.
\item Your group will have three playing cards to represent the three doors (two two's -- the 
goats, and an ace -- the car).  
\item {\bf Round 1}:  Repeat 20 times and record the results -- how many times
the contestant won the car and how many times the contestant won a goat:``Monty'' shuffles 
the cards, deals them out {\it face down}, and notes 
which cards are in which locations.  The ``contestant'' makes the initial choice but
leaves the card face down, Monty
reveals a ``goat'' (not the card chosen by the ``contestant''), the contestant \emph{keeps the 
original choice} and reveals it. 
\item {\bf Round 2}:  Repeat 20 times and record the results -- how many times
the contestant won the car and how many times the contestant won a goat: ``Monty'' shuffles 
the cards, deals them out {\it face down}, and notes 
which cards are in which locations.  The ``contestant'' makes the initial choice but
leaves the card face down, Monty
reveals a ``goat'' (not the card chosen by the ``contestant''), the contestant \emph{switches 
to the other unrevealed card} and sees the result.
\item When you have completed 20 iterations in Round 1, then 20 iterations
in Round 2, we'll combine results from all of the groups and try to understand
what happened. 
\item If ``Monty'' forgets which cards are the goats and reveals the wrong card, 
or if the rules above are not followed in some way, discard the results of that
iteration and \emph{try again}.
\item If you want to switch roles between the two rounds, go ahead.  ``Monty''
needs to pay more careful attention to what's going on so that a correct card
is revealed in each repetition(!)
\end{itemize}
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