If you select an answer to this question at random from the 5 choices below (*), what is the probability that you will be correct?

A: 20%

B: 40%

C: 0%

D: 20%

E: none of the above

(*) uniform probability distribution, "none of the above" includes "the question makes no sense"

added later: It is important that A and D are both 20% for the paradox to work. But assume that we change D to e.g. 30%, this would significantly change the puzzle; but would it be less paradoxical?

## 9 comments:

The answer is "none of the above" so there is a 20% probability of it being chosen. So I think there are two ways to think about it. I can think the two 20% answers are also both correct and so the probability becomes 60% leading to a contradiction and therefore the probability as being undefined. Or, I could think that the only correct answer is "none of the above" which only has a 20% probability of occurring and if one of the 20% answers are chosen they are incorrect answers and don't affect anything so the answer is 20%.

If "none of the above" is correct it has 20% of being chosen, which means it is incorrect, because A and D would be correct.

But A and D would be chosen with 2:5 = 40% probability, so B would be correct, but if B is correct it would only be chosen with 20% probability, so it is false.

So this leaves us with C , if no answer can be chosen correctly, but ...

>> So this leaves us with C , if no answer can be chosen correctly, but ...

I assume since that again leads to a contradiction that you like my first answer of the probability being undefined. It's a good brain teaser!

The "paradox" hangs on the self-referentiality of "this question," but just calling a sentence a question doesn't make it one - or does it?

I don't think the question is the problem. Imagine that we leave the question the same, but change option A and D from 20% to 30% or something else.

Then answer E: none of the above (which includes "this question makes no sense") would be the correct choice and there would be no problem.

>> but just calling a sentence a question doesn't make it one - or does it?

The sentence is definitely a question, but it's a question that doesn't make sense because it is self referential as you point out. According to Wolfgang's rules then the answer is E "none of the above." I disagree with Wolfgang that you can get to a 40% answer from there. If E is correct, then A and D become correct too and the answer is 60% and not 40%. Anyway, regardless of whether I am correct about that or not, since a contradiction of sorts obviously arises I think you could say the probability is undefined. I think you could also use a kind of theory of types though and say that if E is correct then A and D cannot occur so the only correct answer is E and the probability remains at 20%.

>> a question that doesn't make sense

CIP, Lee,

I think it is interesting to consider different options:

replacing D e.g. with 33% instead of 20% etc.

It seems to me that an option 33% or 67% etc. would be clearly irrelevant.

On the other hand 20%, 40%, 60%, 0%, 100% seem to be much more appropriate ...

So if the question and the whole setup is nonsensical, then why is there this difference?

And it would seem that A to E includes all possible answers (because of "none of the above").

Lee,

>> If E is correct, then A and D become correct too

If A and D are correct, then E (none of the above) cannot be correct,

because A and D are above E ... 8-)

>> On the other hand 20%, 40%, 60%, 0%, 100% seem to be much more appropriate

You have a random process so it seems kind of natural to me that multiples of 1/N would be more appropriate.

I think you can probably find a way to get a paradox using your question for any N > 2 using multiples of 1/N. If N = 2 though it is a little tricky. Then you have "none of the above" and 1/2. If you say "none of the above" you end up in an infinite loop. If you say 1/2, you don't end up in a loop, but as far as I can tell there is no particularly logical reason to say 1/2 to start with.

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