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MathGroup Archive 1996

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Monty Hall

  • To: mathgroup at smc.vnet.net
  • Subject: [mg3788] Monty Hall
  • From: Don Piele <piele at cs.uwp.edu>
  • Date: Sun, 21 Apr 1996 23:24:51 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

Monty Hall Problem (Source -  Marilyn vos Savant, Parade Magazine,
Sept 9, 1990)

Monty Hall was the host of the daytime T.V. show "Let's Make A Deal"
that  ran from 1963 to 1990.  One of games he made popular was to
have a contestant  pick a curtain out of three possible curtains
shown on stage. Behind one of  the curtains was a new car, but
behind the other two was a booby prize -- for example a cow.

If you are allowed to only pick one door, your chances of winning
the car are clearly 1/3. So to make it a bit more interesting, Monty
Hall would tempt the contestant in the following way. Let's say they
pick curtain number 1. Before the curtain was opened, he would
reveal what was behind one of the curtains not picked  -- of course
a curtain with a cow behind it,  never the car. Let's say he shows
off the cow behind curtain number 3. Now Monty Hall would allow the
contestant the option of changing their mind and switching from
curtain number 1 to curtain number 2.

The question is: Are they better off switching curtains or staying
put?

When this problem first appeared in Parade Magazine,  many readers,
including mathematicians and statisticians, argured as follows:
"When Monty Hall opened a door and showed you one of the cows, there
are two unopened doors behind which one car and one cow are hidden.
Therefore it does not matter if you change or not, since the chance
of winning the car is 1/2 in both cases."

As is well known by now, the probability of winning if you always switch is
2/3. 

Problem for "Mathematica Peals: Problems and Solutions  (MiER Vol 5 
No 1.)

What happens when we extend this problem to include  N curtains,
behind which we have placed P prizes (cars) and N-P cows (one behind
each curtain), and we repeat the game R times? Create a function
called montyHall[N,P,R] which will simulate the game R times and
compare the chances of winning a prize by following the two
strategies used above: 1) never switch; or  2) always switch. We
assume of course that all curtains are equally likely to hide a
prize and equally likely to be chosen. 

Sample:

montyHall[4,1,10]

Car   Cow   Cow   Cow

Cow   Car   Cow   Cow

Cow   Cow   Car   Cow

Cow   Car   Cow   Cow

Car   Cow   Cow   Cow

Cow   Car   Cow   Cow

Cow   Car   Cow   Cow

Cow   Cow   Car   Cow

Cow   Cow   Cow   Car

Cow   Cow   Car   Cow
                                                           1
Never Switch: Relative Freq ={0.2, 0.4, 0.3, 0.1}, Average -
                                                           4
                                                     	       3
Always Switch: Relative Freq = {0.4, 0.3, 0.35, 0.45}, Average -
                                                               8


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