Re^8: Marilyn Vos Savant's Monty Hall problem

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

Comment on Re^8: Marilyn Vos Savant's Monty Hall problem

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Re^9: Marilyn Vos Savant's Monty Hall problem by tilly (Archbishop) on Aug 24, 2004 at 21:54 UTC
What can I say? If you analyze a probability problem start to finish and add in the assumption that the outcome you saw is the only possible outcome that could have happened, then you'll usually get wrong answers. That isn't a matter of opinion - that's mathematical fact. In case you're curious, here's how the formal approach goes. You start with a priori estimates of various scenarios. You run the experiment and see what happened. You then compute the conditional probabilities of the scenarios of interest given what just happened using the well-known formula from Mr. Bayes, `P(A\|B)=P(A and B)/P(B)`. In English that says, "The probability of A given that B happened is the initial probability that A and B both happened divided by the initial probability that B happened." That gives you the right numbers every time. Any other approach can only be right when it winds up agreeing with that one. Look elsewhere in this thread for the problem that was misstated in tye's textbook about how likely the other child was a daughter. And try to understand the full (if informal) analysis that I gave. You should see how a correct analysis has to consider what might have happened (but didn't). Furthermore you'll see how very subtle differences in the stated problem change what was possible, and result in very different estimates. (Note that I didn't pull out the formal machinery. I would have needed to if I wanted to show how to adjust the answer to account for the fact that about 51% of children are male.) If you still think that I'm wrong, well that is your perogative. I've spent all of the energy on this thread that I'm willing to spend. I'm not here to present a course on elementary probability theory. You're not here to take one. So I'll let it drop here.	[reply] [d/l]

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Re^9: Marilyn Vos Savant's Monty Hall problem
by tilly (Archbishop) on Aug 24, 2004 at 21:54 UTC

If you analyze a probability problem start to finish and add in the assumption that the outcome you saw is the only possible outcome that could have happened, then you'll usually get wrong answers. That isn't a matter of opinion - that's mathematical fact.

In case you're curious, here's how the formal approach goes. You start with a priori estimates of various scenarios. You run the experiment and see what happened. You then compute the conditional probabilities of the scenarios of interest given what just happened using the well-known formula from Mr. Bayes, P(A|B)=P(A and B)/P(B). In English that says, "The probability of A given that B happened is the initial probability that A and B both happened divided by the initial probability that B happened." That gives you the right numbers every time. Any other approach can only be right when it winds up agreeing with that one.

Look elsewhere in this thread for the problem that was misstated in tye's textbook about how likely the other child was a daughter. And try to understand the full (if informal) analysis that I gave. You should see how a correct analysis has to consider what might have happened (but didn't). Furthermore you'll see how very subtle differences in the stated problem change what was possible, and result in very different estimates. (Note that I didn't pull out the formal machinery. I would have needed to if I wanted to show how to adjust the answer to account for the fact that about 51% of children are male.)

If you still think that I'm wrong, well that is your perogative. I've spent all of the energy on this thread that I'm willing to spend. I'm not here to present a course on elementary probability theory. You're not here to take one. So I'll let it drop here.

[reply]
[d/l]