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Twist on Monty Hall


Date: 03/31/2001 at 17:27:39
From: rick singh
Subject: Twist on Monty Hall

Here is a twist on the classic Monty Hall problem. 

There are three doors. Each door contains a prize. Your goal is to 
maximize the probability of getting the best of the three prizes.  
You can open as many of the doors as you like, but you have to stick 
with the last door that you open. When you open a door, you do not 
know whether or not it is the best. 

Should you open only one door, or should you open one and then open 
the second and stick with the second - or should you open all three 
and stick with the third?

I understand Monty Hall, but this one has elements I can't figure out.  
Thanks for your help.


Date: 04/01/2001 at 16:41:37
From: Doctor Schwa
Subject: Re: Twist on Monty Hall

Hi Rick,

The classic problem can be found in the Dr. Math FAQ:

  The Monty Hall Problem
  http://mathforum.org/dr.math/faq/faq.monty.hall.html   

Yours is a really interesting type of problem! I first heard it in the 
context of a princess choosing among a hundred suitors. At any time 
she can accept the proposal of the suitor she's seeing, but once she 
rejects one, he won't come back. How can she maximize her chances of 
getting the best one?

The answer, of course, depends on the relative quality of the suitors.
I think it turns out that the best strategy for her is to let the 
first 37 suitors go by, and then if she ever sees a suitor better than 
the best she's seen so far, keep him. Why 37?  Well, that's a long 
story.

Luckily, your problem has only three doors instead of a hundred 
suitors, so it's a bit easier to analyze.

If you keep the first door, you have a 1/3 chance of getting the best
prize.

If you go to the second door, and just keep it no matter what, it
still has a 1/3 chance of being best. Same with the third door.

BUT, when you open the second door, you already have a bit of
information: is the second door better than the first?

If it is better than the first, keep it. If it's worse than the first,
go on to the third and take whatever's there.

This strategy gives you a 50% chance of getting the best prize.
Let's let A stand for the best prize, and B the second best, and
C the worst.  Then the possibilities are

   ABC
   ACB
   BAC
   BCA
   CAB
   CBA

With the strategy described above, you win whenever the second prize
is best (BAC and CAB), but you also win in the case BCA, because
with the second prize worse than the first, you go on to the third
and win.

So, half the time you get the best prize; in two cases out of six,
ACB and CBA, you end up with the middle-quality prize (in ACB,
since the second prize is worse than the first, you try the third;
in CBA, since the second prize was better than the first, you stay
with it).

Only in case ABC will you get stuck with the worst prize.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Probability

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