It's a multiple choice question with 4 choices, stripping the choices A,B,C, & D of their values to choose one at random gives you a 25% chance of choosing correctly in any multiple choice question. There are two choices for 25%, you would have a 50% chance of choosing the right answer. So the answer is B.
You can't argue this, B is only one choice of 4, so you still have a 25% chance of choosing it at random. "So wouldn't that make the right answer 25%? That's why it's a paradox?" If the answer were 25%, you would have a 50% chance of choosing it at random. B is still the right answer.
The trick is that the question isn't asking you to actually choose on at random, in order to break out of the circle of fallacies you put yourself into you have to split your imagined situation into two dimensions so that you are a spectator watching another person answer the question at random. What is the chance they will answer correctly? Since it is initially 25% by default for any multiple choice, they have two choices for 25%, you now know that they have a 50% chance of answering correctly, thus YOUR answer is 50% while THEIRS is 25%. Any other speculation beyond this is redundant and will lead you to the same conclusion, answer B and continue with the test.
Actually my good sir, by having two 25% answers, you then can eliminate one of them (assuming both are correct to select). This creates a 1/3 chance of getting it right at random, which is 33.333...% therefore none of these are correct, and it's a 0% chance.
Sorry man. There is a slight problem with that argument. It says that "if you choose a question at random, what is the chance that you will be correct?" This would mean that the correct answer is 25% by default. But since there are two answers of 25% that makes 50% the correct answer right? But if 50% is the correct answer, then there is only one 50%. That makes 25% the correct answer. But there are two 25%'s. That makes 50% the right answer. But there is only one 50%. That means 25% has to be the correct answer. And so on and so on.
That is what makes splitting the imagined situation into two dimensions, in which you are the spectator of yourself answering this question at random, very crucial.
You have to envision the continued circular paradox until you reach a point where you realize, "Oh **** it actually is possible to stop here!". You have to take logic and try to throw it an entirely different direction than the one life requires it to go for the paradox to finally reach a conclusion.
Logic cannot be changed. That is why it is logic. You cannot stop at any point and say it is the end because it is a circle. It has no ends. You can say it ends here, but that will never make it true. My conclusion remains the same.
However, there are two choices for 25%. Therefore, if 25% is right, and there are two options for 25%, 50% is right. Both 25% and 50% cannot be right, so 25% cannot be right.
If the answer is 50%, then there are two possible correct answers, meaning there is another answer besides 50% that is correct. There cannot be two correct answers at the same time that are different, so 60% is left.
60% = 2.4/4. This means 2.4 of the answers are correct, which makes no sense.
Okay so actually, it's like this: If we are allowed to make the assumption that the answers A) B) C) D) are the only responses to be picked in a random then the "chance" of picking the right answer is 0% because the correct answer would be 'does not exist' because there is no logical basis to make the assumption that one of the four answers is correct. In reality it's a question hidden behind a misleading question. Daniel Kahneman and Amos Tversky earned the Nobel prize in 2002 for their models showing that intuitive reasoning is flawed in predictable ways and this is a prime example.
It's not a paradox at all, the answer is 50%, because its a 1/4th chance so 25%. Except that 25% is given twice. Therefor the answer is 50%. no paradox involved. Just some tricky word play
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Then the answer would be 50% percent, which only appears once, so the chance would be again 25%, but 25 percent appears twice, so it would be 50% again...
"All possible outcomes are simultaneously correct if unobserved." As pointed out by Closetnerd somewhere in the comments. Best answer I have seen so far.