From: mk452a@oktext.sbc.com (Mark Kamphaus)
Subject: Re: Monty Hall Puzzle RIP
Reply-To: mk452a@oktext.UUCP (Mark Kamphaus)
Organization: Southwestern Bell
Date: Mon, 21 Dec 92 16:11:49 GMT

In article <92353.100332SHERE@SLACVM.SLAC.STANFORD.EDU> <SHERE@SLACVM.SLAC.STANFORD.EDU> writes:
>Alright, I've had just about all bickering I can stand on this
>Monty Hall drivel and I'm going to put it to bed.

[some psudeo-number crunching and other verbage deleted]
>... Now everybody, repeat after me; half the time
>we stay with our original choice we win, half the time we change
>our minds we win, half the time we win, PERIOD!  You have no
>problems with the fact that if Monty reveals what is behind one of
>the doors and then lets you choose between the two remaining that
>your chances are fifty-fifty.  Yet this is exactly what's happening
>here!  The only difference is SEMANTICS!  Whether you "change your
>mind" or "stay with your original choice" you are STILL CHOOSING
>BETWEEN ONE OF TWO DOORS.  FIFTY-FIFTY.
>
>Flames and arguments expected, oh well.
>
> Lee

Lee, you solved the wrong problem.  The key to the Monty Hall problem
is that as it is stated, he must always open an empty door.  This means
that additional information is given by his choice.  He doesn't randomly
open a door to reveal that it is vacant.  In our problem, he must 
use his knowledge to open the vacant door and leave the other one 
closed.  The importance of this can be seen if we try to use the
information to "cheat" in the game.  Assume that you are the contestant
but prior to the show, you caught a glimpse behind the first door
and it was vacant.  You have two choices.  You can pick doors 2 or 3
and see what happens.  Obviously if Monty opens up remaining door that you
did not pick, you wouldn't change and win (you would use your knowledge that
door #1 was vacant as well as the one he just opened).  But if he opens
door #1, you have no additional information and have a 50-50 chance
if you switch.
The other choice you can make is to pick the door that you know is
vacant!  Since Monty will always open the other vancant door,
you should always switch and win.
Now if you don't get a peek behind any of the doors before you
begin, you should still try to use the same strategy.
You should try to pick a door that is vacant and switch
after monty shows you the other vacant one.  Because
you have a 2/3 chance of picking a vacant door initially,
and the cheating strategy guarantees a win P=1, you
chance of winning with this strategy is 2/3*1=2/3.
If this is still difficult, imagine 1000 doors.  
You pick one (initial chance 999/1000 being vacant)
and Monty open up 998 vacant ones. (Remember, the chance
that these doors are vacant is 100% because monty
knows which door contains the prize and must avoid it.)
There are now two doors.  The one you picked, and the one
Monty didn't open.  Since the chance of yours being vacant
hasn't changed since you picked it, you should switch to
the door having the 1/1000 chance of being empty.
And if that reasoning doesn't work, the last bit of
rationale says that there are by definition always
going to be 2 out 3 doors vacant.  If the Game was that
you could pick two doors (one would always be vacant)
or just one, which would you pick?  Two doors hopefully.
After having picked the two doors, if Monty opens one and
it is  vacant, would you switch?  No.  You knew that at least
one of the doors that you picked was going to be vacant.
But you also knew that you had a 2/3 chance of winning
when you picked two doors.  Monty didn't change your
chances with his opening of the door;  You knew a door
was vacant when you picked them, Monty has only shown you
which one it was. So you don't switch and you will win
2/3 of the time.  In our Game, you pick two doors
for Monty (by default when you choose the one
for yourself).  You know that at least one is vacant
Monte shows it to you and then asks if you want
to trade positions.  If you switch, that puts you in the
same position of having chosen two doors initially, even
though a vacant one has been revealed.  You have a 2/3
chance to win.  Hope this helps.


mark
Newsgroups: rec.puzzles