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.
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.
The whole thing is basically the Monty Hall problem.
How is it related to Monty Hall?
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.
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.