My chemistry prof would tell of a student who needed to get just one question right on the multiple-choice final to pass the course. He walked in, picked 'C' for all questions, and walked out.
I am, however, under the impression that if what you want to do (as here) is to minimize the probability that you get completely shafted by having had the test-setter choose answers complementary to yours, you should choose yours in a way that makes that probability low. Random choices are a very effective way of achieving that. [EDITED to add: not because "a random key fits a random lock" but rather the reverse: with non-negligible probability[1] the test-setter is not choosing at random but in some relatively low-Kolmogorov-complexity way, and you want to keep away from those parts of the outcome space.] Of course, having a really good model of the test-setter's decision process would be even better, but if you had that you'd just use it to ace the test.
[1] I initially missed out the word "probability". I edited it in. Sorry.
To whoever's downvoting Eliezer's comment above (I know at least two people have):
If you think he was stupidly wrong to issue the original challenge, downvote that. If you think he was right to issue the original challenge and stupidly wrong to back down when I argued, downvote me since presumably I'm even wronger. But what the hell sense does it make to downvote someone for being prepared to change his mind in the face of disagreement?
Incidentally #2: For the multiple-choice test, even better than choosing random answers is to choose random answers and them check them for low Kolmogorov complexity (in so far as that's possible; there are some theorems restricting it) and generate new random answers if the results are bad. You could turn this into a not-at-all-random algorithm that performs even better, given sufficient (vast) computing power: enumerate, and execute, all "cheap enough" computations that produce sets of answers; use this to put some suitable probability distribution on answer-sets (so that answers generated by cheaper computations are more probable, and then (deterministically) choose your answers to minimize the probability of failure. This is the kind of thing Eliezer has in mind when he claims that every randomized algorithm can be beaten by a derandomized one.
If I only need to get one question right it doesn't take many questions before (3/4)^n is smaller than the chance the professor, for whatever reason, avoided C.
I disagree. I'd put that probability at 0 after n=7.
In fact, according to the recent reddit thread, some test designers are encouraged to enforce equal proportions of each option. While stupid, it lowers the threshold to n=3 or so.
Here's a variant on this that is actually true: I took Professor DiCenzo's "History of Japan" course at Oberlin College, in we he assigned several 5-page papers. There were a variety of topics to choose from (or make up your own); one of those topics was "Zen". Professor DiCenzo was famous for saying "The first student who chose the 'Zen' topic and turned in five blank pages got an 'A'. You will get an 'F' -- please try to use words."
No correct answers were 'C'.