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## #1 2006-03-09 16:13:09

fgarb
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### Cars and Goats

This is from an old American Game Show. The puzzle is a little complicated to describe, but I'll do my best.

Contestants are given three doors to choose from. Behind two doors are goats, and behind the other door is the grand prize, a nice new car. The contestant gets to pick one door, and whatever is behind it he gets to keep. The catch is, after the contestant picks the door, the host opens one of the doors the contestant did not pick to reveal a goat. The host then gives the contestant a chance to change his mind.

As an example, say the contestant picks door 3. The host then reveals that door 1 has a goat behind it. The contestant then has the option to switch guesses to door 2 or stick with door 3.

The question is, is it more likely for the contestant to win the car if he sticks with his original door, or if he changes doors, or are the odds equal either way?

## #2 2006-03-09 16:59:35

Ricky
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### Re: Cars and Goats

It took me writing a computer program to simulate it before I accepted the reasoning behind it as valid.

Great puzzle.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #3 2006-03-10 04:31:24

mathsyperson
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### Re: Cars and Goats

Ah, the good old Monty Hall problem again. This has already been discussed at length here.

Why did the vector cross the road?
It wanted to be normal.

## #4 2006-03-10 05:29:54

Ricky
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### Re: Cars and Goats

I love how all arguments seem to die in the face of a good simulation.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #5 2006-03-10 13:52:10

fgarb
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### Re: Cars and Goats

That was a fairly extensive debate! I actually went through a similar process myself when I first got this puzzle - I couldn't believe the answer was really 1 in 3, so I ran a simulation of it. Of course, as usually happens for me, the process of writing the code showed me the flaw in my reasoning, and by the time I was done with it, running it was only academic. I tend to not understand things as clearly as I think I do the first time around, but something about writing the process down clearly in a way that a mentally challenged two year old (or a computer) could understand usually shows me that I'm not as smart as I think I am.

By the way, I found this online in regards to this puzzle. Aparently a full blown math professor made a fool out of himself over this, which makes me feel less bad

"When Marilyn vos Savant quoted this puzzle in the US a few years ago, she received over 10,000 letters mostly telling her she was wrong.

One was from Robert Sachs, a professor of mathematics at George Mason University in Fairfax, Va. who said "As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and, in the future, being more careful."

However a week later and Dr. Sachs wrote her another letter telling her that "after removing my foot from my mouth I'm now eating humble pie. I vowed as penance to answer all the people who wrote to castigate me. It's been an intense professional embarrassment."

Full text at http://www.grand-illusions.com/monty.htm