I mean, I agree that it’s a defective prompt in the pedagogical sense. I would protest as a student. But I’m gonna carry on with the analysis.

Btw, I accidentally clicked smthg on your post when I went to reply. Dunno what it was or what it did. If I somehow flagged you or smthg, I am profusely sorry.

Nobody got flagged. Just checked.

Thank you, sir/madam.

See, there could be a point to that. Suppose it were done this way:

16. ...
17. Do not select an answer to this question.
18. ...

Then the instructor could assess a heavy penalty on anyone who bubbled in any answer to (17). It would be a check on students answering randomly.

I mean, without even reading the questions.

That is interesting.

I see you creating an entire narrative, a fantasy world, to try and facilitate your position. It is on a test, it is being graded, there is an instructor, there are students, etc. . .

Not only are you making a ton of assumptions about the nature of all these fictional story elements, but you are taking the OP out of context.

What is see is people doing what people tend to do: Unnecessarily over complicate things. It is why the Law of Parsimony exist, to help mitigate such observational bias.

I would call that more of a guideline, second to the guideline that your theory should be empirically adequate. That said, I don’t consider this a question of theory choice. It’s a straightforward exercise in chance with a self-referential but non-circular twist. I quite enjoyed it.

By the way, I appreciate your dialing down the vitriol. I am happy to debate this as friends.

People want to wrap this question up in a nice bow, with everything in its place, that's why some think 0% should be C), as they think it makes it nice and tidy.

Some want an answer, even though the question, which does have an answer, can't be answered. That is where the 33% comes in, they need it to have a best answer. It is human nature, to try and put everything in its place even if it has no place; we are the diligent organizers of an endless chaos.

However, to get to those points they must redefine the question; they must change the question. As if they are trying to avoid the original question, because it messes with their human nature.

The question as it is in the OP is the best version if it, as it is nonsensical chaos on more levels than one, which is the point. It is a reminder that these systems we have built, like mathematics, are not perfect and they only work up to a point. It is a reminder that for all our great insights, our understanding of even our own concepts is incomplete.

Here are two more versions.

1. Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct?

A) 25%
B) 50%
C) 60%
D) 25%

Only when you go to the answer sheet, you see (I'm putting the letters where there would be bubbles in labeled columns)

1. A. B. C

Wait, so is D part of the sample space or not, since it turns out I can't "choose" it?

2. Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct?

A) 25%
B) 50%
C) 60%
D)

Now what? There's "D", but can I answer "D"? If I assert that D is the answer, what would I be asserting?

Which brings me to this point: a question and answer pair should be recastible as an argument or at least an assertion. If it can't, then it's either inconsistent or a contradiction.

I get the point you're making about mathematics, I think. If I drag in some context, it's too show that mathematics doesn't stand naked on its own, but relies on a broader conception of rationality. That conception can be seen at work in the way we talk to each other, and in the way we make tests. It's those conventions your question violates, and that's why it's a conundrum.

I actually like the puzzle, else I wouldn't spend my time trying to figure out how it works and why it resists solution.

The question as it is in the OP is the best version if it — Jeremiah

No it isn't. The answer to the version in the OP is 0%, which just isn't offered as an option. You might as well ask:

Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct?

A) 1%
B) 2%
C) 3%
D) 4%

Or:

What is the capital city of France?

A) London
B) Washington D.C.
C) Berlin
D) Ottawa

The paradox works best when 0% is offered as an option.

If 0% was in the OP we would have lost that entire part of the discussion.

I think it is time for a new conundrum, something easier. I have one in mind, I just hope it is not too easy.

I have no idea what you mean. But the version in the OP isn't a conundrum at all. It just doesn't list a valid answer.

I've just been reminded of this one which I found amusing along similar lines.

What is the right answer to this question?
Is it...

A) b
B) c
C) d
D) a

@Jeremiah

If you revise the parameters to say the teacher picks one and only one of A or D to be true regardless of the content of the answer, then I will accept your analysis. Then the answer is simply 25% bc, even though it’s listed twice, it doesn’t matter what the answer choices actually say. We can observe that 50% of the students get the semantically correct answer, but we don’t go in circles because we stipulated that the criterion of correctness has nothing to do with the actual question, so the actual frequency of correct answers is irrelevant to the teacher’s designated answer choice. I appreciate the conundrum that, on some interpretations, the meaning of the question is part of the format of the question.

Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct? — Jeremiah

Saying they each have a 33% chance is suggesting they all have the same probability of being chosen by the chance event, which is not true; 25 has double the chance of being selected than the other values. — Jeremiah

Sorry if I'm still trying to understand this problem, you seem to have moved on to another thread.

I think (as other participants here are seeing also), the "chance of being correct " in the first quote above, and the "probability of being chosen [by chance event]" imply two different commands. I say "imply" although it is clearly obvious. In the second quote from your post, you are referring to 'what's a chance that each choice will be randomly selected. This is a good question if you're doing it in a blind selection. But the choices are out there listed as multiple choice. So, the question as I see it is, how many times a 25% occurs? That's 50% of the 4 choices. (As you correctly identified).
Then, there is a second question -- what is the chance you will be correct?
So, let's say 25% is the correct answer -- and it occurs twice in the multiple choice, giving it 50% occurrence rate, does it also mean that the chance of us being correct is 50%?
I'm choosing from 3 unique answers, not 4.

My question is really for this:

I'd argue the frequency of the answer's appearance in the multiple choice list should not affect the chance of the answer being correct. — Fool

This is what's at stake.

An answer is correct if and only if its value matches the chance that an answer with that value will be selected. — Michael

I think that formulation is incorrect, because if this truth condition yields "true" for more than one value, the chance to be correct <i>overall</i> is greater than for any of the individual values.

Take for example:

a) 25 %
b) 50 %
c) 50 %
d) 60 %

A has a value with a probability of 25 % to be chosen, so it's correct. B and C both have both have a value that has a chance of 50 % to be chosen, so they're correct, too. But that would render the chance to be correct overall at 75 % and according to the problem's formulation, none of them would be correct. But if none of them is correct, then the way we arrived at the correctness of the individual values isn't valid, as it doesn't address the problem.

I don't even know how to formulate this problem in mathematical terms. I don't understand the truth condition.

A has a value with a probability of 25 % to be chosen, so it's correct. B and C both have both have a value that has a chance of 50 % to be chosen, so they're correct, too. But that would render the chance to be correct overall at 75 % and according to the problem's formulation, none of them would be correct. But if none of them is correct, then the way we arrived at the correctness of the individual values isn't valid, as it doesn't address the problem. — Dawnstorm

Good point.

the way we arrived at the correctness of the individual values isn't valid, as it doesn't address the problem. — Dawnstorm

True.

I think you just can't admit that you were wrong. — Jeremiah

That is not philosophy, not by a long shot and if that is the standard that passes on these forums, then I have to question if I belong here at all. — Jeremiah

Because my arumgent is sound, besides I hate it when everyone sits around agreeing with each other, it is incredibly unproductive. — Jeremiah

This popped up this morning:
https://www.gocomics.com/fminus/2018/05/23

Thought it expressed my takeaway on this topic.

And I am sure that was your only take away.