Probabilistic liar paradox

This example is similar to the classic liar paradox, only recast in terms of probability.

The question

If you choose an answer to this question at random, what is the chance that the answer you choose is correct?

• A) 25%
• B) 50%
• C) 0%
• D) 25%

The original liar paradox

The example suffers from the classic problem of questions that refer to themselves. That is why we mentioned the liar paradox above, which can go like this: “This sentence is false.” The paradox is that if the sentence really is false, then it is telling the truth. But if the sentence is telling the truth, then it cannot be true, because that is exactly what it claims!

The solution

In the solution we assume a uniform probability distribution, so each answer can be chosen at random with probability 25%.

• Suppose that answer A is correct. What matters here is that answer D contains the same percentage, so the probability that we choose the answer 25% is 50%. So neither answer A nor answer D can be correct.
• The probability that we choose answer B is 25%. So answer B cannot be correct either, because it says that we have a 50% chance of choosing the correct answer.
• The probability that we choose answer C is 25%. So answer C cannot be correct either, because it says that we have a 0% chance of choosing the correct answer.

As you can see, none of the answers is correct. So what chance do we have of choosing the correct answer? If none of the answers is correct, then we have a 0% chance of choosing the correct answer. But that contradicts the fact that we actually have a 25% probability of choosing the answer that says we have a 0% chance.

So the question has no solution. It is a nice paradox, just like the classic liar paradox.