Question: If you choose an answer to this question at random what is the chance that you will be correct?
Answer: A premise of the show is that each question has one correct answer listed. Therefore the two 25% answers are not viable answers. With there being only two viable answers, you have a 50% chance of being correct if you guess between them.
If you choose any answer, not just viable ones, then there are an infinite number of choices and the correct answer would be zero.
It says you choose an answer at random, though, not choose from viable answers.
I think this is just a way for the studio execs to keep their million dollars.
At the face value, 1 out of 4 options gives you 25% chance. If we assume 0,25 (regardless of repetition) and 50 are the choices, then you have 33% chance. If you eliminate 25% as the choice because it appears twice in the answer, then the chance of 1/2, i.e., 50%. If it is assumed that you are able to eliminate the two 25% choices, then your thought process is not random. Based on this, if we can assume you can use logic correctly and there is an answer to the question, then we can assume you will reach the right conclusion through correct use of logic. Therefore, the chances are 100% that you will get to that answer. If you assume infinite number of possibilities for an answer, that would make the answer 0%. However, as soon as you choose 0%, this will contradict (because you gave the correct answer despite 0% chance) with itself. Thus, the question turns in to “what are the chances that you will have the same line of thought about what is really being asked with the person posing this question and getting to the right answer”.
If you choose 0% it doesn’t contradict, because you didn’t pick it at random, it’s the correct answer. But if you answered at random you would have essentially 0% chance of being correct.