The Multiple Choice Paradox

I believe most of you won't like to see the following question on your exam paper.

If you choose an answer to this question at random,
what is the probability that you will be correct?
A) 25%
B) 50%
C) 0%
D) 25%

This is actually a very hard question. To fully understand the question we need to make some assumptions and see what we can infer from them. Let's start from an "obvious" one.

Assumption 1: this question has a unique correct answer.

Assumption 1 can be deducted from the type of question, i.e. "multiple choice", if no more information is provided. Generally speaking, when there are four choices and one is picked up randomly, the probability of getting a correct answer is 25%. However, this is only true when four choices are different. In this specific question, both A) and D) are 25%, so the probability of choosing 25% (the original correct value) in a random choice is actually 50%. As a result, B) seems to be a more reasonable answer.

But we can't choose B). The reason is that if B) is correct, then the probability of choose B) (the current correct answer) from four choices is 25%, which contradicts with B) itself. Actually we can prove that Assumption 1 restricts the answer must be 25%, so neither B) nor C) is a correct choice, so we can only select from A) or D).

Based on the assumption, the correct answer should be either A) or D), but not both. Since A) and D) have the same value, how can we choose one and ignore the other? Here is where random plays. The professor who offers this question may have a preference each time. Each student who takes this exam also has his/her own preference. When their preferences match, the student can get credits. This is reasonable, but it can rarely happen in real world. Those students who figure out the correct number but pick the wrong answer have sufficient reasons to accuse and fire their crazy professor.

After the whole reasoning, you may realize it is not a good idea to stick on assumption 1. By removing this restriction we may find out a better solution. So let's move to assumption 2.

Assumption 2: this question has more than one correct answer.

If we adopt the assumption 2, it immediately implies that we should choose both A) and D), and the problem is solved. But... if you read the question more carefully, you will find that the probability is calculated when student can choose only one answer. So in this case the probability goes to 0%, then C) is the correct answer, which contradicts with Assumption 2.

I guess you now understand that there are some serious problems in the question. Maybe it doesn't have a correct answer? Let's try to explore that option.

Assumption 3: this question has no correct answer.

Unfortunately it doesn't help either. When there is no correct answer, whatever answer selected is wrong, so the probability of choosing a correct answer is 0%, which is a valid choice and contradicts with the assumption.

After exploring all possible cases, we reach the conclusion: in normal situation there is no good way to deal with this question. Even though the question is not so bad as a paradox, since we still have one way to go and rely on luck, I think it can be called "the multiple choice paradox". Its logic is a little more complicated than those famous paradoxes, but the construction is the same: they all use self reference to create impossible sets.

Before closing the discussion, I have more words with the only possible but not reasonable way, e.g. the correct answer will be one of A) and D), and students rely on luck to get credits. If we regard each exam as an observation, before a student (observer) receives his/her paper, the correct answer is an event that occupies the whole event space. It can be viewed as the superposition of both A) and D). At the exact time of looking at the paper, the state of event collapses and become a distinct value (either A) or D)). The distinct value can change in different exams but the state of superposition is the same all the time. If in some cases no student ever gets the paper nor the score, then the answer will remain in the superposition state forever because no observation is made. This idea is worth mentioning because scientists use that to describe quantum physics! As we can see, when observation is very limited, we use terms like superposition to describe the situation because there is no good alternative. The underlying mechanism can be arbitrary but no one could reveal it!