But that is a big assumption. In analyzing probabilities you cannot just work with what has happened, you have to work with what could have happened instead. A probability problem is never fully specified until it includes both knowledge of what did happen and what might have happened.

This is where I think we're in disagreement... Not with your statement, because I agree with it if you're trying to predict future behavior.

But that's not what we're trying to do in this case. We're not 'about to go on the show'... and we're not 'about to begin the prize round'.

The problem states that we're *already* in the prize round, chosen a door, and the host has revealed a goat door. That's where we *start*

So we have to calculate the probabilities based on that set of variables. You can't examine the possibility that "The host might not have opened a goat door first" because this question *started* with that stated.

Basically, I guess I'm stating that I agree with you that if we're talking about all possible future behavior, then we need to consider that the host might just open the prize door... or not offer a choice... etc...

But for this specific problem, we already find ourselves faced with the host opening a goat door and offering a switch, so to decide *from here* what to do, we have to assume these as fixed variables in our analysis.

You're beginning before the game started... I'm beginning from the point where two actions have already occured... Which changes things.

It's like turning on the TV and finding a car race in it's last lap and wondering what odds are that the car in the back of the pack could win... You would base your conclusion on how many times you've seen the back car win from the last lap in the past... You wouldn't consider the possibility that he *might* have been in the front or the middle, because he isn't...

If you were going to analyze the probability of his winning the *next* race, all that comes into play... but to analyze the probability that he's going to win *this* race, then you discard those options that can no longer occur.

That's how I see the Monty Hall problem. You're considering the probabilities based on *any* version of the game... I'm considering the probabilities of this *one* game, where the goat door is already open, and a choice has been offered.

And, since someone's mentioned 'arguing' with you... for the record, I've enjoyed this discussion, regardless of the outcome of it, or whether we ever agree on it :)

Trek

In reply to Re^8: Marilyn Vos Savant's Monty Hall problem by TrekNoid
in thread Marilyn Vos Savant's Monty Hall problem by mutated

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