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Re: [Phys-l] Monty Hall problem



Tim Folkerts wrote:

There is nothing in [the original problem statement] that explicitly indicates any sort of pattern. I only see that in ONE INSTANCE the host offered this choice. Just saying "DOES" is ambiguous in this case. It allows both: 'Host DOES THIS TIME open a goat door and DOES THIS TIME offer a switch' and 'Host DOES ALWAYS open a goat door and DOES ALWAYS offer a switch'

Precisely. This is the one and only source of contention here, the one that makes Marilyn's answer strictly wrong.

Notice, however, that the problem statement does not EVEN say "does." Indeed, everything about the problem statement--its use of the present tense, the way it (needlessly) specifies the precise door numbers, the way it makes "you" and your need to make a (single) choice the center of attention--strongly suggests that this is an "instance" that may or may not, therefore, characterize what would happen in any other instance.

Thus, to answer the question with confidence, one absolutely must either get additional information. Otherwise one is reduced to making an assumption. I have consistently agreed that the "always does" assumption is the simplest, *perhaps* even "the most obvious" assumption, one that leads to a very interesting and, for many, counterintuitive result. But is it "the most reasonable"? That seems to me to be at best an undecidable question.

As John D and I have pointed out, if the host is not constrained always to "show a goat and offer a switch" and is acting as an adversary, then the only reason he is offering you the chance to switch is because you have already chosen the car. In that case, you are guaranteed to lose if you accept his offer. That might even be "the most reasonable" assumption.

And this is hardly an academic question. Quite the contrary! The problem statement puts *you* in the situation and asks what *you* would do. Does anyone think they would automatically make the assumption that the host is NOT an adversary or that he ALWAYS "shows a goat and offers a switch" if they were in that situation? As I previously reported, the simple fact is that Monty Hall did NOT always "show a goat and offer a switch."

As I mentioned before, Marilyn's "visualization" actually puts an even finer point on all of the above. She wrote:

Suppose there are a million doors, and you pick door #1. Then the host, who knows what's behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You'd switch to that door pretty fast, wouldn't you?


I'm not at all sure. First, if there are a million doors, then the top prize is likely to be EXTREMELY valuable, so much so that the show's finances may very well hinge on NOT giving it away. Second of all, if the host opened 999,998 of the doors to show me that that top prize was not behind any of them and then offered me the chance to switch, THE uppermost thought in my mind would be "Why on Earth did he do that?!?" Finally, the fact that I chose door #1, only tends increase my suspicions about all of the above because I can imagine that they might have put the prize behind door #1 on the assumption that nobody would make such a "silly choice."

Finally, as I said at the very beginning of this thread, under the "simplest assumption" that the host ALWAYS "shows a goat and offers a switch," the most compelling explanation by far IMO is simply to consider the expected results of 3000 instances of the game: 1000 times the car ends up behind the door you originally selected and the other 2000 times it ends up behind the one you are ALWAYS allowed to switch to.

John Mallinckrodt
Cal Poly Pomona