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.
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.