What, exactly, would the guard would say in different situations? Using the standard, utterly unrealistic, interpretation of probability problems like this, the guard is supposed to say this:
VM: If you are on Vulcan, you are in the Mountain
VD: I have nothing to tell you.
EM: If you are on Vulcan, you are in the Mountain
ED: If you are on Vulcan, you are in the Mountain
in which case the probability is 1/3. But I have a hard time believing that the guard is willing to talk to you here, but wouldn't be willing to talk if you were in the Vulcan Desert.
Since the guard refused to talk at first, but then told you something later, it seems pretty clear that they're trying to help you out. The most obvious way for them to communicate to you where you are is like this:
VM: If you are on Vulcan, you are in the Mountain
VD: If you are on Vulcan, you are in the Desert
EM: If you are on Earth, you are in the Mountain
ED: If you are on Earth, you are in the Desert
But there are other possibilities. Perhaps there's a policy of executing guards that reveal information about where you are, so the guard wants plausible deniability by lying to you:
VM: If you are on Vulcan, you are in the Desert
VD: If you are on Vulcan, you are in the Mountain
EM: If you are on Earth, you are in the Desert
ED: If you are on Earth, you are in the Mountain
It seems that you've ruled that out in the problem statement, though.
Altogether, as Dacyn says, "it depends on what you know about the psychology of the guard."
Somewhere in Rationality, there's a post about this.
"Somewhere in Rationality, there's a post about this." - do you have any idea where such a post is?
I believe I was thinking of this one:
https://www.lesswrong.com/posts/f6ZLxEWaankRZ2Crv/probability-is-in-the-mind
If you assume that the guard's probability of making this statement (and only this statement) is the same in all circumstances where the statement is true, then the answer is 1/3. Otherwise, it depends on what you know about the psychology of the guard.
EDIT: This was wrong.
The answer varies with the generating algorithm of the statement the guard makes.
In this example, he told you that you were not in one of the places you're not in (the Vulcan Desert). If he always does this, then the probability is 1/4; if you had been in the Vulcan Desert, he would have told you that you were not in one of the other three.
If he always tells you whether or not you're in the Vulcan Desert, then once you hear him say you're not your probability of being in the Vulcan Mountain is 1/3.
There are N options, the host removes one option with the (presumed) constraint that they only can remove one of n-1 options which aren't true. What's the remaining probability? Monty has a further constraint that he can't remove the one you picked, but it's very a similar underlying calculation.
Also identical in that common formulations are ambiguous about the rules for guard/monty to eliminate an option, leading to irksome and unnecessary controversy.
In this example, he told you that you were not in one of the places you're not in (the Vulcan Desert). If he always does this, then the probability is 1/4; if you had been in the Vulcan Desert, he would have told you that you were not in one of the other three.
That can't be right -- if the probability of being in the Vulcan Mountain is 1/4 and the probability of being in the Vulcan Desert (per the guard) is 0, then the probability of being on Earth would have to be 3/4.
P(vulcan mountain | you're not in vulcan desert) = 1/3
P(vulcan mountain | guard says "you're not in vulcan desert") = P(guard says "you're not in vulcan desert" | vulcan mountain) * P(vulcan mountain) / P(guard says "you're not in vulcan desert") = ((1/3) * (1/4)) / ((3/4) * (1/3)) = 1/3
Woops, you're right; nevermind! There are algorithms that do give different results, such as justinpombrio mentions above.
Here's an interesting riddle that is more complicated than it looks:
You wake up locked inside a room with no window. You know your captors have four facilities in the following locations: a Vulcan Mountain, a Vulcan Desert, an Earth Mountain, an Earth Desert. They flip a coin to decide which planet to send you to, then they flip a coin to decide which facility on that planet to use.
Wanting to know where you are, you try to get some information out of the guard. He refuses at first, but eventually he offers the following: "If you are on Vulcan, you are in the Mountain". What are the chances that you are in the Vulcan Mountain facility?
(You can assume that the guard is telling the truth and not trying to intentionally manipulate the situation)