An Excellent Statistics Question

Now here's a truly excellent statistics question.

Those of you at Caltech who also took Alan Hájek's "Philosophy of Probability" course will know exactly what this question reminds me of. :)

Hajek's test was the best test ever when it comes to exam-taking paranoia: It was multiple choice. For each question, instead of picking a correct answer, we assigned a probability of correctness to each possibility, and our score on each question was determined by a formula that included a logarithm. Woe to anybody who assigned probability 0 of correctness to a choice that turned out to be correct, because then one got a grade of -infinity in the course. The result of this was to ask oneself nervously on every question: Am I truly sure that I am 100% confident that this answer isn't right? (Not only that, but exactly how confident am I on every possible answer for every question?)

[The scoring methodology was announced before the exam when I took the course, so I computed beforehand exactly what I should place instead of '0' on the questions in which I was maximally confident that some choice wasn't correct.]

(Tip of the cap to Tammy Porter and Danny Suiza.)