On Wed, Oct 28, 2009 at 4:09 PM, Dan Asimov <dasimov@earthlink.net> wrote:
The reasoning is simple: After the initial guess, the two remaining doors play identical roles with respect to what the player knows. The player knows that at least one of them hides a goat. So when a goat is shown, that gives the player no new information about the original guess.
This is true regardless of what algorithm may be used to choose the goat door, as long as the player doesn't know the algorithm.
I agree that if some statement is made that is equivalent to "at least one of the two doors contains a goat", such as the revealing of a goat by someone who knows which door is which and always shows a goat, then there's no new information. But if someone randomly opens a door, and it contains a goat, then you have information like "door B contains a goat" ... quite a different thing. The problem is what you can deduce from a statement like "door B contains a goat". If you know the algorithm by which door B was chosen, then I think we all agree that there are many possibilities for how that impacts the probability of door A containing a goat (still 1/3, or 1/2, or perhaps you can even get 0 or 1 from some of the situations discussed earlier). But what should you do if you don't know the algorithm, and don't know the knowledge of the door opener? It seems to me that rather than assuming "door opener knows what is in all the doors and always reveals a goat", the more null-hypothesis-like choice would be to think "door opener chooses a random door which this time happened to have a goat", in which case you get a probability of 1/2 instead of 1/3. Maybe this all boils down to what the definition of probability really is. Is it, or can it be, a measure of your ignorance? If it isn't, then how can ignorance of the algorithm mean that the probability can't change from 1/3? (What about a Bayesian approach with various priors for possible door-opening strategies?) Maybe this all boils down to the question of what constitutes information in this context. I agree that the statement "at least one door contains a goat" has no new information. But in practice the door opener opens a particular door, and maybe that action communicates some information: "door B contains a goat" is not the same statement. --Joshua Zucker