I know I'm late to this thread, but something just reminded me of this question and I thought I'd see if the folks here at xkcd had discussed it yet.
The thing I think is interesting about this question is the unspoken assumptions people make when it is posed to them.
Most people (not necessarily here, but everywhere else I've seen this) seem to assume that the answers listed are the only possible answers, i.e. we're not allowed to pick other answers. I think that whether or not that assumption is true, 314man got the best* (*see below for details) answer in the first reply: the odds of randomly guessing the correct answer to this question are 0%, because...
- If the assumption (and another to be specified below) is true and we are only allowed to choose between those four, then we will randomly choose 25% 50% of the time, 50% 25% of the time, and 60% 25% of the time, and thus never choose the correct answer, making the correct answer 0%, which is not listed and so will be chosen from among those listed 0% of the time, as it says.
- If the assumption is false (but the other below is true), then we have an infinite range of unlisted answers to choose from and the odds of randomly choosing any one of them is infinitesimal, i.e. zero, including the odds of choosing zero, which means the only answer which could possibly be correct is zero, even though we will choose it on average zero percent of the time.
But the really interesting thing that I think everyone overlooks is the assumption that a random distribution is a uniform distribution. There are plenty of random distributions which are nonuniform. We just have no reason to think that whatever random choice process we use to select an answer to this question will be nonuniform in the distribution of its choices. But (given the first assumption considered above, that we have to pick between these four answers), if something about the universe was really frickin' strange, and any random process used to answer this question randomly picked answer C three out of five times on average, then C could logically be correct: there would be a 60% chance of picking answer C, which is what C says.
Of course, empirically, epistemically, the odds of the universe being biased in a way such as to make any random process used to answer this question pick answer C 60% of the time are really, really, really, really, really, really... really slim. So the technically correct full answer, assuming we are only allowed to pick between these four answers, is "mostly probably zero, but maybe, though most improbably, 60%". On the slim epistemic odds that the universe is so weird, we've accounted for that in our answer, and it is correct; and the rest of the time, our answer is not among the available choices, and so the odds of us picking it from them at random are zero, and it is also correct.
Now, where it gets really weird is with a modified question: what if instead of answer C saying 60%, it said 0%? Then the odds of choosing correctly between the four answers via a uniformly random process would still be zero... but the odds of choosing 0% as an answer would be 25%. Now we've got a logical contradiction on our hands... unless the universe is really weird, and random processes randomly never select answer C to this modified question. So does that mean that posing this question forces the universe to be weirdly biased, on pain of logical contradiction?
Of course, if we allow the possibility of such weird cosmic bias, then it could just as well be the case that B gets randomly selected 50% of the time and A, C, and D each 16.66_% of the time, so B) 50% is correct; or that A and D each get selected 12.5% of the time, and B and C each 37.5% of the time, making both A) 25% and D) 25% correct. In other words if the universe wants to be biased, it can make whatever answer it wants to any form of such a question correct.