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Behind three doors are two goats and a new car (each behind it's own door). You like cars more than goats, so let's assume you are out to win the car. You have guessed what's behind door number one. Before Monty shows you what's behind door number one, he gonna open up one of the other doors to show you where one of the goats is. It's door number 3. Baaah-baah. Okay, now Monty poses a question to you. Do you want to still stay with what's behind door number one, or do you want to switch to door number two? Do you switch? Does it make any difference with your chances of winning the car? Give your answer and reasons for defending it. And don't go searching around for answers on the web! Hash out your answers here.

I think I would stick with door #1, although the chances are still 50/50 regardless of whether I stay with door #1 or switch to door #2. Sometimes I can just be stubborn.

It really depends on my gut instinct and feel at that time... Looking at it one way, there is a 50:50 chance of winning once one door has been opened. But looking at it another way, your original choice had a 1/3 chance of being right and a 2/3 chance of being wrong. If you stick with it, you will win if your original chance was right (33%) but if you switch, you will win if your original choice was wrong (66%)...

It really doen't matter if you choose door # 1 or door # 2; you still have a 1/3 chance of winning the car. You now have 2/3 of the doors remaing, and a 1/2 chance between those two doors of finding the car. 1/2 x 2/3 = 1/3. Basically, changing your guess does not increase you chance of winning (it still remains at 33% overall, but 50% for each doors #1 and #2 once door #3 is revealed). This question was beaten to death about 10 years ago in Marilyn vos Savant's weekly column. The debate went on between her and her readers for months as to whether or not the chances of winning increased if you changed doors. She said they didn't (for the reason I outlined above), but readers disagreed. Still, Door #1 and Door #2 each have identical chances of winning at all times. (33% each overall, 50% each once door #3 is revealed). Since they both have identical chances of winning, unless you can find some other way to make your decision (such as Monty Hall's reactions, the day of the week, the aligment of the planets, etc.), either door is as good as the other. As for me, I'd take the goat because I already have a car.

Piscis Babelis est parvus, flavus, et hiridicus, et est probabiliter insolitissima raritas in toto mundo.

Joe McGuire
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Joined: Mar 19, 2001
Posts: 293

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Are you all so sure it doesn't make any difference to switch?

This question was beaten to death about 10 years ago in Marilyn vos Savant's weekly column. The debate went on between her and her readers for months as to whether or not the chances of winning increased if you changed doors. She said they didn't (for the reason I outlined above), but readers disagreed.

Joel, I think you got it backwards - Marilyn vos Savant advocated that one should indeed switch! (Source: Copyright 1991 The New York Times Company, The New York Times, July 21, 1991, Sunday, Late Edition By switching, you increase your odds from 1/3 to 2/3. But one doesn't have to be Marilyn vos Savant to see what's really going on. It's a standard probability question covered in most introductory statistics classes. It is important to remember that Monty does indeed know where the car is, and is not going to show you what's behind door number one, whether there's a car behind door number 1 (which is 1/3 of the time, or whether there's a goat behind door number 1 (which is 2/3 of the time). He's always going to show you the other goat. You're getting extra information about where the car is 2/3 of the time. Still not convinced? Check out the simulation results done by the Mathematics Department of the University of San Diego; or better yet, write your own simulation in java!

The thing that Marilyn vos Savant missed here is that it makes a difference whether (a) Monty always opens a second door after your guess and offers you the chance to switch, or (b) Monty chooses whether or not to offer this option. This is not clearly specified in the problem, either as Joe put it, or as MvS did. MvS implicitly assumed case (a), and answered correctly for that case. But some of her readers were assuming case (b), which is far more complex and ultimately unresolvable, and MvS was unable to detect the difference in their arguments - rather than listening long enough to understand, she assumed that everyone who wrote her was just doing it the same incorrect way. Many were, but still... It's worth noting that the real-life Monty Hall did in fact choose whether or not to open one of the doors, and employed a variety of other mindgames to steer the contestant away from the correct choice, whatever it was. (Or did he steer them towards the correct choice sometimes, for better ratings?) So you could never really be sure what rules he was actually operating under. [This message has been edited by Jim Yingst (edited November 05, 2001).]

My error! I was just recalling from back when I was a young'un, and way back then I ddn't exactly follow the arguments for either side...I was just trying to reconstruct the arguments from memory.

Originally posted by Jim Yingst: It's worth noting that the real-life Monty Hall did in fact choose whether or not to open one of the doors, and employed a variety of other mindgames to steer the contestant away from the correct choice, whatever it was. (Or did he steer them towards the correct choice sometimes, for better ratings?) So you could never really be sure what rules he was actually operating under.

This is why I said that the aligment of the planets was a good way to decide. And I'll still take the goat.