Let's hear your logical proof for dropping a datum.

Not one person gave a valid reason for dropping a datum.

You don't just remove observations to fit your answer, that is called data hacking and as far as I am concerned, we are analyzing a data set with four observations, so you need a damn good reason to just drop one.

It's exactly the same as everyone else here. The question asks for the selection of an "answer" at random, there are three possible "answers" (25%, 50% and 60%) one of which, it is implied by the format, is correct. The fact that I've been provided with two different ways of indicating to the questioner that I've chosen 25% from the three available options, is odd, but irrelevant to this interpretation.

No one, not one person gave a valid reason for dropping a datum. — Jeremiah

I'll ask again, what would define a valid reason? Everyone here has given you what they believe to be a reason. I can guarantee you that this forum is made up at least from some of your epistemic peers (other people of your intellect) . If they're wrong about their reasons being valid then it follows that it must be possible for someone of your intellect to be wrong about this issue. So how do you know it isn't you?

Implied?

That has observational bias all over it. That is the difference that you are not seeing, I am not reinterpreting the question.

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

No one's "re-interpreting" the question more than any other. We're all interpreting it one way on our first pass, maybe seeing another interpretation on second reading. What makes your first interpretation more right than anyone else's first, second or third attempt?

As I said, your argument "my argument is sound... because it is", is not a style of logical analysis I'm familiar with.

Listen, as I told Michael, my entire arumgent is on the table, and I have repeated myself many times already. It is all there just go read it. Also don't confuse the word logical with the word reasonable.

Imagine the question were written like this:

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%

Answer: ...

You have to write on the dotted line the answer. How many different things can we write on the dotted line? I say just three: 25%, 50%, or 60%. If there are only three possible answers then my sample space for the random selection has only three elements.

I already know your argument, Michael. Unless you are repeating yourself for someone else?

My position has not changed, despite your renewed effort.

If you choose an answer to this question at random — Jeremiah

Given a multiple choice test with a fixed format, say, every question having four possible answers, there are two ways to choose randomly:

(1) reading the questions and the answers first;
(2) not reading them and just bubbling in something on the answer sheet.

By design these are equivalent on multiple choice tests because answers are never deliberately duplicated. If use method (2), it won't even matter to you that this question directs you to choose randomly -- that's what you're doing already. People tend to use method (1) in part so they can throw out "(a) a fish" as a possible answer to "What is 6 x 7?" and then randomly select among the other three.

I claim all this is relevant because you rely on our expectations about how multiple choice tests work but then screw with them. It is literally a trick question, just not the sort that's typical on these tests.

Suppose this was a real test, and the duplication was a mistake. Then the instructor would discover that this question is broken. Admittedly it's broken in a really unusual way, but the result is that no student can give a correct answer. As far as that goes, it's no different from a typo in which the correct answer was supposed to be "(d) 33%" but was printed as "(d) 55%".

And again, all of this is lost on the student who just randomly bubbled in A or B or C or D without even reading the broken question. To him, the question is just

Blah blah blah
A something
B something
C something
D something

Do you have a proof that he is Doing It Wrong™?

The argument shouldn’t be about the answer choice sample space. There are 4 elements and, when you de-dupe, the frequency of ‘25%’ has to be 50%. But given that weighting, we want to know how often the chosen answer is the correct answer to the question. That implies a second chance event, the (formally) unpredictable outcome of what the correct answer actually is. That sample space only has 3 elements.

The second 25% is clearly intentional and not a typo.

The second 25% is obviously a typo that leaves no correct answer as an option.

To the student who just bubbles away, there's no difference between a broken question and a question he just didn't luckily answer correctly.

To the test designer, a duplicated answer is always a mistake that can only help such students.

It was not a mistake the question has two 25% and one 50%, with four possible answers. That very much seems to "imply" a sample space of four.

This is not a test and clearly the second 25% is meant to be there.

Anyone else want to back the horse that the second 25% was put there by mistake? Or can we all at least agree the question was purposely designed with it?

I find it interesting that the three sample space notion requires more hoops to jump through than the four sample space. That's why people feel the need for these out of context examples; they have to redefine the question.

Can we say Occam's Razor?

I’m not sympathetic to the “broken question” view. I assume there’s a correct answer, and then I assume the answer choice selection is random, as stipulated. The meanings of the question and answer choices don’t affect random selection. Of course, you revise to 0% after noticing the formally correct answer isn’t given, but this defect doesn’t have anything to do with the duplicative answer choice. You could imagine a teacher doing that and still successfully grading the test.

You might say the two-event analysis looks more baroque, but it’s also more adequate.

You could imagine a teacher doing that and still successfully grading the test. — Fool

By marking all answers as wrong?

Yeah. I didn’t say it would be fair. It would be coherent. The test also wouldn’t have any pedagogical use, but that’s a practical problem.

That's the definition of a broken question.

In that case, broken questions can still be coherent.

But not answered correctly. And the question is about answering correctly, so it's not like I'm forcing my own preconceptions on it.

Anyone else want to back the horse that the second 25% was put there by mistake? Or can we all at least agree the question was purposely designed with it? — Jeremiah

I see two possibilities:

(1) I assume you've made a mistake, because the question cannot be answered as posed (Principle of Charity and all that);
(2) You disabuse me of (1) by demonstrating the correct answer.

I’m construing it as a true/false check on a send/receive transaction. The test sends a prompt and sets a “correct answer” variable on the back end. The test-taker sends a response. The program checks to see whether the response equals the correct answer. You could run that program, no problem. The frequency would be 0%, but the program runs.

I’m construing it as a true/false check on a send/receive transaction. — Fool

But we're not asked what the chance is that we've succeeded in answering; we're asked what the chance is that we've answered correctly.

I don’t see the distinction. If I had to choose what I’m trying to analyze, it would be the latter. I agree, we don’t care how many students successfully mark an answer. How many of them mark an answer that is the same as the correct answer to the question? Well, the correct answer isn’t listed in the prompt, so none of them sends it back.