That extra information is an assumption that is supposed to be based on your knowledge of how game shows work. In fact based on Monty's knowledge and motivations it is possible that you want to switch (Monty knows where the car is and always shows a goat), it doesn't matter (Monty doesn't know where the car is), or you should stick with your choice (Monty knows where the car is and is trying to keep you from getting it).

I've never seen the Monty Hall problem presented with that stipulation explicit.

Here's the exact text of the original question, posed to Marilyn in her column in parade magazine:

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 number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?

http://www.fortunecity.com/victorian/vangogh/111/9.htm

The stipulation that the host knows what's behind the doors, and always opens a door with a goat is a given for this problem.

In fact, the additional information that it was 'Monty Hall' came later... which messed the whole thing up because they interviewed him, and he stated that he sometimes opened the prize door right away... and then everyone forgot the stipulations of the original question

I actually remember when this happened, because my college statistics professor was *consumed* with proving Marilyn wrong, and ended up conceding she was correct only after writing a program to do 10,000 permutations that bore her true

Trek

And that statement, while better than how it is presented the vast majority of the time, still isn't sufficient to make the answer unambiguous. There is still the question of Monty's motivations which could make it either adviseable to switch (up to 2/3 odds of winning) or to stay (up to 100% chance of winning).

An interesting follow-up on Marilyn. Later she tackled a restricted version of the problem that I presented at Spooky math problem (the restriction being that the two envelopes hold money, one has twice the money of the other) and correctly analyzed an argument for whether you should always switch. But she incorrectly analyzed whether you could do better than even odds. I know a couple of probability theorists who pointed out her mistake to her, but she never admitted to her mistake.

Make of that what you will.

I've never seen the Monty Hall problem presented with that stipulation explicit.

Funny, I've always seen it explicitly stated that the host knows and will not open the door with the car. If it wasn't this way, he'd open the door with the car 1/3 of the time. A quick google search seems to confirm this, though I only checked the first 4 links that weren't applets. The extra information Mr. Hall provides is what makes the puzzle worth puzzling ;)

Update: strike off the cuff response... By the way (after a quick read and next to no analysis) the spooky math problem looks more like a divide-by-an-infinite-number problem. The probability that you pick a number between two numbers in an infinite continuum is undefined.

Please read Re^7: Marilyn Vos Savant's Monty Hall problem carefully. You'll find that it is not enough to say how much knowledge Mr. Hall has - to make the answer unambiguous you also need to specify how Mr. Hall will use that knowledge.

On the spooky math problem, there is no divide by an infinite number problem, but there is a lot of subtle behaviour with infinity lurking around. I strongly recommend against coming to any opinion without thinking it through very carefully. It took me about a week to really understand what was happening when I first ran into the problem. (It didn't help that Laurie let me get myself well and truly convinced that there was no way that you could beat 50% before he gave me the answer and told me to figure out for myself why it worked...)

Given any pair of numbers in the envelopes, and given the distribution from which you pick a random number, your probability of winning is well-defined and will be bigger than 50%. Depending on those three factors, the actual odds can be anywhere between (but not including) 50% and 100%. But you are not provided with sufficient information to figure out what the odds are.

For what little it's worth...

People seem to be arguing with tilly about this, but this has been my experience too. I have seen this problem published at least a dozen times; I have never seen the assumptions (e.g. that Monty knows what is behind the doors, etc.) stated clearly before.

buckaduck