Perhaps the ultimate reduction of this line of questions:
If you chose an answer to this question at random, would it be the correct answer?
A. Yes
B. No
First - you choose an answer at random - so every choice is equally probable. Every answer will be chosen 25% of the time, assuming pure randomness. But it is impossible to choose answers by randomness and be right 25% of the time. You will be right 50% of the time under that strategy.
So - you cannot choose an answer whose value is the number of times you are correct. It’s a word play paradox, like the sentence “this sentence is false” or “I am lying”.
I don’t think there is a correct answer it’s a null set. Or something.
The beginning premise is that you are choosing an answer to this question at random. If you choose any answer other than by random selection, the premise of the question no longer holds. So the answer is to just pick an answer at random, and that’s the correct answer because you picked it randomly.
So I picked 0% by choosing the option that represented the quadrant the second hand of my watch was in when I looked at it.
That seems to be the only way to break the paradox. The need to randomly select the answer overcomes choosing anything else, because it’s part of the premise. If you don’t do it, there is no question.
How would you determine the correct answer without trying? And if by “could” you don’t mean “how many will you get right if you write them right now?” It still not a “correct answer that you should deduce or already know”, it’s a survey question with all the issues of “correctness” survey questions have.
I’m going with C. 50%
I’m pretty sure this is not correct.
If I’m following your logic, what you’re saying is the equivalent of this:
If you choose an answer to this question at random, what is the chance that you will be correct?
A. 25%
B. Joe Biden
C. Blue
D. Snow
Three of the four answers aren’t numbers; if you randomly chose any of those answers they would have to be wrong. They wouldn’t magically become the right answer just because you chose them at random like the question asked.
What question?
I am reminded of an anecdote about Paul Dirac. At one of his lectures a member of the audience said 'Professor, I do not understand the derivation of the formula on the top left of the blackboard".
There was a long silence. Eventually the moderator said, Professor Dirac, are you going to answer the question?
Dirac replied: “That was not a question, it was a statement. Next question, please”.
The question is “what is the chance you will be correct?” i.e. What are the odds you will be correct?
Odds imply total randomness, die are not loaded, etc. So the odds of selecting any particular answer is 0.25 or 1/4. No fudging that answer. it’s not “what did you choose this time?” It’s “how many times in 100 or 1000 or 1,000,000 tries will you get the correct answer?”
But since both A and D are 0.25, the odds of selecting 0.25 are 0.50; you cannot choose an answer of 0.25 and be correct.
“But wait!” you say. “That means if I select C 50% I will be right.” True, but then you are right 25% of the time, because you will only select C 25% of the time, and therefore C would not be correct. All you’ve done is create a circular logical paradox.
The correct answer should be 25% and you should (must) be able to pick that answer exactly 1/4 of the time. this is impossible. Therefore a valid answer is not listed.
But if no valid answer is listed, then you have a 0% chance of picking the correct answer. And you have a 25% chance of picking 0% as your answer. So 25% is the correct answer and you have a 50% chance of picking 25% as your answer. So 50% is the correct answer and-
And that’s when the thread blows up and Kirk regains control of the Enterprise.
Indeed. Not everything that looks like a valid question actually is one.
Rather than a paradox, isn’t this sort of a version of category error? The first part of the question is putting forward a hypothetical about performing an action that has no correct or incorrect “answer”; the second part is asking the probability you will be correct in your performance of that action.
It sounds like a real question because both parts of it a referencing mathematical things. But the question in the OP makes no more sense than asking: If you hop up and down on one foot for two minutes, what is the chance you will be correct?.
Because I have encountered multiple choice questions where there was no “correct” answer–perhaps due to an error in constructing the question and answers, perhaps due to an error in keying–I must respond with “I don’t know.”
What is the correct value of X, where X is a non-zero integer?
A. 0.25
B. 0
C. 0.50
D. 0.25
Just because you can ask a question and follow it up with a selection of answers doesn’t mean I must conclude that the correct answer is included in the selection.
I feel that while this particular question might be a paradox, it’s not a categorical issue.
For example, if they made one small change:
If you choose an answer to this question at random, what is the chance that you will be correct?
A. 25%
B. 0%
C. 50%
D. 75%
The problem would have a simple objectively correct answer.
This shows that you can have a self-referential question that is logically consistent and solvable.
My favourite “Portia’s caskets” question from Smullyan’s book, as told by Martin Gardner:
Gold casket: “The portrait is not in here.”
Silver casket: “Exactly one of these statements is true.”
Where is Portia’s portrait?
Answer: you do not know. In Gardner’s version, the portrait happened to be in the silver casket.
It would be helpful if you gave the set-up. In the original Shakespeare, there were three caskets (a third was made of lead). Is there a third one in this riddle or only two? Do we know that there is exactly one portrait and that it is inside one of the caskets?
Based on what you’ve given us and assuming there is one portrait and it’s inside a casket, then it’s in the gold one. The statement on the gold casket saying it isn’t inside is false. The statement on the silver casket is true which makes it the one true statement out of the two given.
The portrait being inside the silver casket would create a paradox. If it’s inside the silver casket, it can’t be inside the gold one. So the statement on the gold casket is true. In order for the statement on the silver casket to be true, either both statements must be false or both must be true. If the statement on the gold casket is true, they can’t both be false. But if both statements were true, then the statement on the silver casket must be true. So there could be only one true statement. This paradox eliminates the silver casket as the location of the portrait.
Or you can just write whatever you please on caskets, and then put the portrait inside, oh, say, the silver one.
The set-up (by Smullyan) varies the original problem by changing the inscriptions on the caskets, the number of caskets, the contents of the caskets (there is not always a portrait!) and other information, but in the penultimate problem there are only two caskets, inscribed as in my previous post, inside one of which is Portia’s portrait, and the suitor wants to open the casket containing the portrait.
“The suitor should have realized that without any information given about the truth or falsity of any of the sentences, nor any information given about the relation of their truth values, the sentences could say anything, and the object (portrait or dagger, as the case may be) could be anywhere. Good heavens, I can take any number of caskets that I please and put an object in one of them and then write any inscriptions at all on the lids; these sentences won’t convey any information whatsoever. So Portia was not really lying; all she said was that the object in question was in one of the boxes, and in each case it really was.”
So? 
I was able to solve the situation you described pretty easily (and I explained how I did it). I have a hard time accepting that Martin Gardner got the wrong answer.
To me the question seems meaningless regardless of the answer choices.
In this revised version, what is the objectively correct answer and why?