Then careful analysis proved her correct.
https://en.wikipedia.org/wiki/Monty_Hall_problem#Savant_and_...
[0] It has to be specified that a) Monty knows what's behind which door, and b) he will on purpose always open a door such that there's a goat behind it.
As far as I’ve been able to find the text below would be the original question (“the host, who knows what’s behind the doors, opens”) and answer (“the host, who knows what’s behind the doors and will always avoid the one with the prize, opens”).
Are you referring to that?
https://web.archive.org/web/20130121183432/http://marilynvos...
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? [Craig F. Whitaker - Columbia, Maryland]
Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. 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?
You see that Marilyn's answer contains the clarified correct specification ("the host, who knows what’s behind the doors and will always avoid the one with the prize..."), but the question does not fully specify it.
Thus, ignoring the clarification in the answer (which furthermore pertains to a modified problem), one could interpret the question differently: the host randomly opens a door [0], this specific day it happens to contain a goat, should you switch?
And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory).
[0] Note that specifying that the host knows what's behind what door doesn't help. They could still pick a door randomly.
> And that ambiguity has given rise to so much spilled ink (and, by the way, the misconception that statistic professors don't understand probability theory).
I don't think this stands either - the letters quoted don't say anything about the distinction you're making. Unless they're all fictional or selectively edited, a bunch of PhDs really did get the puzzle wrong.
If you specify the problem carefully, anyone with some training in probability should get the answer right. But clearly back then it was a head scratcher.
I wonder whether there've been other problems like that, or we will encounter similar ones: elementary and easy to understand, yet many people get it confidently wrong, until the correct solution permeates through culture.
I don’t understand what do you mean by that. The clarification was part of the original answer in the context of the original question. The clarification was stated again and again and again. If that’s a “modified problem” so be it but that was the problem being discussed with her readers.
I would say that the whole (short) paragraph is intended to clarify why the answer to the question is "Yes; you should switch."
Maybe it was not a good way to visualize what happened after all. Experience shows that some readers were unable to understand the argument.
However, none of them complained about the explanation being wrong because those were two completely different problems (only in the second one the host will always avoid the door with the prize, apparently).
This is a common way to explain it, but I always found it less intuitive; not sure why.
My go to thought experiment along these lines is rather this:
Adjust the game to where both you and your friend are playing at the same time. You _NEVER_ switch doors, and your friend _ALWAYS_ does, when given the choice by Monty.
Since you never switch, no matter what Monty does you know you're going to win 1/3 of the time. Since you both know Monty is always going to show a "goat" door, ONE of you must win, so your friend MUST WIN ALL THE TIMES YOU DO NOT. Since you win 1/3 of the time, the wins "left" is 2/3 of the time.
You don't need condition (a) here. It's enough to just stipulate that a door with a goat behind it will be opened, however that comes about