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A probabilistic version of the liar paradox.
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Aaron Li's profile photoAnders Aagaard's profile photoKevin Bourrillion's profile photoRoger Craig's profile photo
26 comments
 
my mind got trapped in an infinite loop.
 
Also interesting if the answer choices were: 25, 50, 50, 100.
 
Will it still be a paradox if all outcomes are not equally likely?
 
0% is a correct answer; how is this different from any other multiple-choice question where none of the listed answers are right?
 
+Patrick LaVictoire, it's different because if you pick an answer, you change the answer. So if 0% were an option, and you picked it, the answer would be 100%.
 
One can also close this loophole by replacing 60% with 0% in answer C.
 
+Terence Tao, how would that close the loophole? If you replaced 60% with 0%, the other answers would still not be correct. And if 0% is the correct answer, and you choose 0%, then you are 100% correct, so 0% cannot be the right answer.
 
I'm with +Patrick LaVictoire. We can easily determine that none of the answers listed is correct. Therefore we know that if your method is to choose (at random) from the options given, you will never get the correct answer. So the probability is 0%. If I had a nickel for every time I had to answer a multiple choice question on a test where none of the answers was right.....
 
you definitely need to close the loop hole, as the self referential question does have a fixed point (0%), also the fix of replacing choice C from 60% to 0% is more elegant, as currently 60% seems to be a junk number there without any purpose.
 
+Kevin Bourrillion , +Patrick LaVictoire It's still a non-standard question because the answers (as a list) are self-referential.

I would prefer the phrasing "Which answer, if chosen randomly, is the probability of choosing itself?" since "correctness" does not seem to apply to situations where more than one answer works equally well (in math).

This is akin to the question: what is the longest a list of numbers (x_1, ... , x_n) can be such that P(choosing x_i) = x_i ? The same as count(x_i) = x_i * n; so we could ask for integers, count(y_i) = y_i (think y_i = x_i * n) ? We can choose something arbitrarily long: (1, 2, 2, 3, 3, 3, <4 4's>, etc).
 
the answer is 42 as it is for everything... sheesh. piece of cake :P
 
Here I'm trying to do a semi-formal proof. Please correct me if I'm wrong in any step.
[Assumption1] When identifying a paradox, one normally iterates through all possible assignment of a variable in the statement.
If each and every of the assignment leads to contradiction, then that state is a paradox.
[Fact1] Notice that the number of correct choices can only take one of the 5 value in the set {0,1,2,3,4}
[Let] the number of correct choices be Y.
[Let] the answer to the question be X. (the percent chance of randomly answering the question correct.)
[Fact2] X = Y*25
[Fact3] the choices are (25,50,60,25)
[Try Y=0] X should be 0 by Fact2. and the answer 0 didn't appear in the list.
[Try Y=1] X should be 25 by Fact2. But the answer 25 appear 2 times, hence Y = 2. contradiction.
[Try Y=2] X should be 50 by Fact2. But the answer 50 appear 1 time, hence Y = 1. contradiction.
[Try Y=3] X should be 75 by Fact2. But the answer 75 appear 0 time, hence Y = 0. contradiction.
[Try Y=4] X should be 100 by Fact2. But the answer 100 appear 0 time, hence Y = 0. contradiction.
[Conclusion]
There is one possible answer, 0.
[Take home]
I think we can generalise the choice. Would someone else do it? thanks for you attention.
 
Some thought of mine, using standard probability theory, gives answer 0%

P(Answer Correct | Answer randomly chosen from below) = P(Answer Correct, Answer randomly chosen from below) / P(Answer randomly chosen from below) = P(Answer Correct, Answer randomly chosen from below) / P(Answer randomly chosen from below) = P(A,B) / P(B) = P(B in A) / P(B in S) [where S = the set of all answers (everything in the universe)] = [can be intuitively shown by drawing some diagrams] P(S in A) = 0
 
I spot a hidden assumption I made with the previous proof.
[Assumption2] There is ONE correct answer.
[Note] Sure the proof includes the possibility that there is NO correct answer too, in that case the proof will conclude a paradox.
The importance of this assumption becomes clear the proof is applied to choices (25,75,75,75)
where picking Y=0, 1, or 3 doesn't lead to contradiction.
 
The question is not a mathematically valid question (first order logic and set theory does not allow self-referring). Still, it's fun to play around with. :)
 
What is the probability of answering THIS question correctly? ..... I always get this right (and wrong)
 
Oh, and if you find a correct answer, then maybe you can also answer THIS?
 
0% seems like the correct answer! BUT as Tao mentioned this is really a probabilistic version of liar paradox. The problem is not really a well-posed probability problem. Its self referencing sentence is not a LOGICAL statement. This will be more clear if we replace 60% with 0%. The contradiction would be more clear in this case but even having 60%, the problem IS still ill-posed although less apparent and it "looks logical" in this case.
 
The self-referential character of the question does not preclude its being logically well-formed. There are all sorts of logical frameworks that permit this sort of self-reference. Notably, any first-order logical system containing a bit of arithmetic allows for the "arithmetization of syntax" that Gödel invented to prove his famous incompleteness theorems, wherein the formulas of the language of the system are coded as natural numbers, and this permits a rather robust sort of self-reference. There are other perfectly sound logical systems that use less oblique devices such as the down arrow of Jon Barwise and +John Etchemendy's book The Liar that basically formalizes the "this sentence" operator of English.

My knowledge of probability theory is terrible, so with some trepidation I'll put forward my understanding of the issue. Suppose first we reframe the question (equivalently, I think) thus: If you have a 4-sided die whose sides are marked "A", "B", "C", and "D", what is the probability that, when you throw it, the letter that comes up labels the correct answer to this question? It seems to me that we get a paradox only on the assumption that 

(*) The correct answer to this question is among the options listed.

On that assumption, exactly one of the listed answers, 25%, 50%, and 60%, is correct. (There are obviously so-called "grounding" issues here, but never mind.) Suppose then the answer is 25%. But, as the probability you will roll an A or a D is 50%, the probability you will get the right answer is 50%. So the answer can't be 25%. So suppose the correct answer is 50%. But the probability you will roll a B is 25%. So 50% can't be the correct answer either. By the same reasoning the correct answer can't be 60%. So, given (*), we have a paradox.

But this "paradox" is easily avoided simply by rejecting the dubious assumption (*) — obviously, there are multiple choice questions that fail to list a correct answer. So it appears to me that all we've got in the problem as stated is such a multiple choice question. But as +Terence Tao notes, a genuine paradox does seem to emerge if we replace answer C with 0%. For, by the reasoning above, the problem as stated DOES have a correct answer, namely, 0% — there is no positive probability that you can roll a correct answer. So, with the change to answer C noted, our possible correct answers are now 0%, 25%, and 50%. By the reasoning above, 25% and 50% cannot be correct. But suppose the correct answer is 0%. As before, we have a 25% chance of rolling a C and, hence, a 25% chance of getting the correct answer. But then 0% ISN'T the correct answer. But this means, once again, that there was no correct answer, i.e., that the probability of rolling a correct answer was 0%. So that IS the correct answer after all.

This paradox (if I've got it right) doesn't rely on the dubious assumption (*) above — we can of course assume that there are multiple choice questions that don't list a correct answer. The paradox here is that, if you assume there is a correct answer, there isn't one; and if you assume there isn't one, there is.
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