For those of you who have never heard of the Monty Hall Problem

For those of you who are familiar with the problem and/or saw the movie 21, pop up some popcorn and just watch until the answer is revealed.

Here goes...

This is an apparently simple problem of statistical and logical analysis and all you have to do is consider a very easily explained problem and answer a rather straightforward question. Here's the setup:

Suppose you're on a game show hosted by famous game show host Monty Hall, and you're given the choice of three doors. Behind one door is a brand new Lamborghini and behind the other two doors are ugly and smelly goats. You win whatever is behind the door you choose. You have no basis for choosing so you go ahead and just pick door No. 1. Monty, who knows what's behind the doors, tries to make things simpler by opening door No. 3, revealing a goat. He then says to you, "Do you want to stay with door No. 1 or do you want to switch to door No. 2?" 

Should you switch?

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Yes, switching doubles your chances of getting the car. Sounds crazy but it's true.

Let's see if others agree. We'll call that your guess.

You have a 50/50 chance at that point....

I would probably stay....but....I don't know, lol

Hopefully, a couple more people will bite, then I'll give you the answer.

Belle...try and work out all of the possible scenarios :)

Ummm....what "all" answers? there's only 2 possible answers....so.....he has a 50 percent chance of getting the right one....so....

I guess I don't see how switching does anything to increase his odds...

Belle, the increase in odds comes from the fact that the host of the show takes away one of the doors that he KNOWS does not have the prize.

Or we can think about it another way, say we have 100 doors and you pick one (1 in 100 chance of winning. Then Monty takes away 98 of those doors since he knows they do not contain the prize. So the two remaining doors are: the one you picked originally and the one Monty left over. The one you picked was picked effectively randomly, so it still has that 1 in 100 chance of winning. The other is effectively a cumulative door (It is the result of the host analyzing and selecting it from the remaining 99 doors) and has a 99 in 100 chance of being the prize door.

This is by far the best explanation I've heard! It makes the problem so much easier to analyze.

Agreed. Far better to visualise and understand. Did you get that from somewhere or did you come up with it yourself Matt? Very nice one.

I don't know if Matt thought of it independently, but among the explanations I remember seeing on Youtube, one of them used the more doors example. However, Matt's exposition helped it make more sense to me.

Also, it's easy for someone to discount the fact that Monty Hall knows what's behind the doors and makes his choice of the door to open based on that information, and think that that fact is irrelevant.

Doubles? no. You go from 1 in 3 to 1 in 2 chance of winning.

Edit: apologies, you are actually correct. your chance goes to 2 in 3 if you switch

Nicolas Taleb (Author of Black Swan which I keep insisting is an amazing read) covers this in "Fooled by randomness" which is also a great read (though less challenging and more fun). He delights in the fact that major players in mathematics and statistics were utterly wrong in their insistance and were outsmarted by a "mere woman" and her weekly article.

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