# Multiple-choice question about the probability of a random answer to itself being correct

I found this math "problem" on the internet, and I'm wondering if it has an answer:

Question: 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%

Does this question have a correct answer?

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23
Yes. This can be answered (correctly or incorrectly) with probability 1. :) – Bill Cook Oct 27 '11 at 22:23

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The answer to the question in the title is currently: Yes, two. – joriki Oct 27 '11 at 22:45

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– user18383 Oct 28 '11 at 2:24

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In the question, I would put the quotes around "math" instead of "problem". It is definitely a problem, but whether it is math is problematic. One issue is whether correctness is computed against the set {a,b,c,d} or {0%, 25%, 50%}. – Ross Millikan Oct 28 '11 at 4:37

6
This should be tagged philosophy. Does anything have an answer? Is there really such a thing as a 'question'? Can we be certain that we really exist? (Answer to previous question: Probably.) – muntoo Oct 28 '11 at 5:01
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## 10 Answers

No, it is not meaningful. 25% is correct iff 50% is correct, and 50% is correct iff 25% is correct, so it can be neither of those two (because if both are correct, the only correct answer could be 75% which is not even an option). But it cannot be 0% either, because then the correct answer would be 25%. So none of the answers are correct, so the answer must be 0%. But then it is 25%. And so forth.

It's a multiple-choice variant (with bells and whistles) of the classical liar paradox, which asks whether the statement

This statement is false.

is true or false. There are various more or less contrived "philosophical" attempts to resolve it, but by far the most common resolution is to deny that the statement means anything in the first place; therefore it is also meaningless to ask for its truth value.

Edited much later to add: There's a variant of this puzzle that's very popular on the internet at the moment, in which answer option (c) is 60% rather than 0%. In this variant it is at least internally consistent to claim that all of the answers are wrong, and so the possibility of getting a right one by choosing randomly is 0%.

Whether this actually resolves the variant puzzle is more a matter of taste and temperament than an objective mathematical question. It is not in general true for self-referencing questions that simply being internally consistent is enough for an answer to be unambiguously right; otherwise the question

Is the correct answer to this question "yes"?

would have two different "right" answers, because "yes" and "no" are both internally consistent. In the 60% variant of the puzzle it is happens that the only internally consistent answer is "0%", but even so one might, as a matter of caution, still deny that such reasoning by elimination is valid for self-referential statements at all. If one adopts this stance, one would still consider the 60% variant meaningless.

One rationale for taking this strict position would be that we don't want to accept reasoning by elimination on

True or false?

• The Great Pumpkin exists.
• Both of these statements are false.

where the only internally consistent resolution is that the first statement is true and the second one is false. However, it appears to be unsound to conclude that the Great Pumpkin exists on the basis simply that the puzzle was posed.

On the other hand, it is difficult to argue that there is no possible principle that will cordon off the Great Pumpkin example as meaningless while still allowing the 60% variant to be meaningful.

In the end, though, these things are more matters of taste and philosophy than they are mathematics. In mathematics we generally prefer to play it safe and completely refuse to work with explicitly self-referential statements. This avoids the risk of paradox, and does not seem to hinder mathematical arguments about the things mathematicians are ordinarily interested in. So whatever one decides to do with the question-about-itself, what one does is not really mathematics.

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7
Liar paradox is also known as the Epimenides paradox after Epimenides of Crete who claimed: "All Cretans are liars". – lewellen Oct 28 '11 at 1:39

2
Bertrand's paradox, the proof of the halting problem, and the proof of Godel's incompleteness theorem all rely on similar self-referencing statements. – BlueRaja - Danny Pflughoeft Oct 28 '11 at 7:29

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@BlueRaja, are you on first names with Bertrand Russell? Usually, "Bertrand's paradox" is about failure to specify a distribution when saying "chose a random chord in a circle". An interesting distinction for Gödel's argument (and the halting problem, which is closely analogous) is that in Gödel's case the self-referential statement doesn't itself know it's being self-referential. It just happens to be. If we do some trivial rewriting on it (such as swapping two conjuncts somewhere) it still means the same but is not exactly self-referential. – Henning Makholm Oct 28 '11 at 10:13
– Did Oct 28 '11 at 13:27

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How dare you question the existence of the Great Pumpkin!? – joriki Nov 1 '11 at 15:30
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The question is underspecified since it doesn't say which distribution is used in choosing an answer at random. Any of the answers could be correct:

If I choose a. with probability 25% and b. with probability 75%, a and d are correct.

If I choose a. with probability 50% and b. with probability 50%, b is correct.

If I choose a. with probability 75% and b. with probability 25%, c is correct.

From the design of the question, it seems that whoever wrote it had in mind a uniform distribution over all four answers, but forgot to specify that. In that case Henning's answer applies.

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9
The writer of the question was definitely a programmer, then. We expect rand() to give numbers that are 'uniformly distributed'... Unlike some mathematicians. (*Rolls eyes.* :)) – muntoo Oct 28 '11 at 5:08

To offer up another perspective on Henning's answer, the question is essentially an elaboration of this (similar) multiple-choice question:

What is the correct answer to this question?

1. Answer (2)
2. Answer (3)
3. Answer (4)
4. Answer (1)

Note that there are some fine puzzles built around variants of the 'self-referential test'; for instance, this simple example:

Each of the following statements is either true or false. Which of them are true and which are false?

1. All of these sentences are false.
2. Exactly 1 of these sentences is true.
3. Exactly 2 of these sentences are true.
4. Exactly 3 of these sentences are true.
5. Exactly 4 of these sentences are true.
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15
I'm probably missing something subtle, but how can the answer be anything other than "Exactly 1 of these sentences is true" being the only true sentence? – fluffy Oct 28 '11 at 0:53

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@fluffy: that's exactly right (though it's often phrased in inverse - 'exactly 4 of these sentences are false', 'exactly 2 of these sentences are false', etc. - which makes it a little trickier). I said it was a simple example. :-) – Steven Stadnicki Oct 28 '11 at 1:36

If there is one right answer to the question, then you will answer this question statistically 25 percent of the time. If 25 percent is the "right" answer, then you actually have two options.

If you have 2 options, then 50 percent is the statistical answer. And if since 50 percent is the only option place to mark down, that means that you will only get this answer right 25 percent of the time because you have a 1 in four chance.

It is impossible without a miracle. Plus, if it is impossible then does that leave the option of 0 open because then there are no right answers? That is saying: "If there are no right answers, this is the right answer." What are you really saying there? Nothing.

I think maybe you can't find out how many answers there are in the first place. There can't be only one. There can't be only two. There can't be three. There can't be four, and therefore one is the right answer? No. Because then you start back at the beginning.

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11
Please don't leave downvotes without providing a comment to explain (see the FAQ). In the present case, this answer is certainly less focussed than some of the other ones, but in my view not bad enough to merit a downvote. – joriki Oct 28 '11 at 0:47

Well, I must be pretty insane if I start competing with these already heavily upvoted answers from high rep users. But although the following solution may sound a little creative or even frivolous, it could easily be the right one. You could say that this solution is reverse engineered, as follows:

The question instructs to only choose one single answer out of four. And assume a uniform distribution, since that is most likely intended, then each answer has a chance of 25% to become chosen.

So the correct answer should be: 25%.

This computes to answer A being correct, as well as answer D. Could that be? Yes, it can. The question does not reveal how many of the four given answers are correct, but since there is one to be picked, assume that at least one of the four answers is correct.

Let's call answer A + answer D the correct answer pair.

Now, there are two possible choices (A or D) that result in 50% of the correct answer (A and D). Secondly, there is 50% chance of picking one (A or D) of two (A and D) out of four (A to D). So whether answer A or answer D is chosen, in either case the probability of being correct (50% × 50%) is 25%, which evaluates true.

Thus, yes, the question has 2 correct answers.

And now I realize that this post is the long version of the by joriki ages ago given comment. ;)

Ok, to be clear about what I mean, I believe the question is a special variant of the following trick question:

What is the color of the car?

1. Black
2. Blue
3. Gray
4. Metallic

As owner of the car I know the correct answer is metallic black. But this would render the question unfair, because it is never possible to give this answer by only selecting one. The difference with the question in question is the equality of both answers to give, which makes it slightly more fair. But since you can select only half of the full solution, the probability is still 25%.

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2
If I'm reading your response correctly, you're both giving meaning to the answers (answers A and D are possibly correct because they are both 25%), and ignoring this meaning (by assuming that one of A or D is the "correct" answer, as in on an answer key or in a computer). This doesn't make much sense... – Michael Boratko Oct 28 '11 at 2:28
@process91 Imagine the answers not being radio buttons as in a conventional multiple choice question, but instead being check boxes: A and D must be checked for the full solution, but you can choose only one. – NGLN Oct 28 '11 at 2:32

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That doesn't seem to be a fair analogy, since "Metallic Black" is not an option, and "Metallic"$\ne$"Black". Based on Joriki's answer above, this doesn't seem to be what he was thinking. – Michael Boratko Oct 28 '11 at 19:13
So, fuzzy logic then? – kinokijuf Jan 23 '12 at 20:30

See problem 2 here for a similar problem that can be solved.

SPOILER: Solution here. Don't look if you want to solve it yourself.

a) can't be the answer because it says b) is correct, but the statement of b) directly contradicts the statement of a)

c) can't be correct, because it means a) and b) are correct but they contradict each other

d) can't be correct. If it were, it would imply a) or b) or c) to be correct. The only possibility left is b), since I have already ruled out a) and c). But, b) contradicts d).

b) can’t be correct for the same reason, basically. If it were true, since a) and c) can’t be true, this would imply d) is correct. But b) contradicts d).

f) can't be correct. If it were, it would imply that e) is also correct, which would contradict the statement of f).

This leaves only e) and none of the statements contradict e) so e) must be the correct answer.

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So, what's wrong with a similar question that actually has a solution? And, where are the coward downvoters with their explanations? Hiding. – Graphth Oct 29 '11 at 1:39

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I didn't downvote, but this doesn't seem to be an answer to the question, and is instead discussing some entirely different question! – ShreevatsaR Oct 30 '11 at 15:57

1
Yes, an entirely different, but very similar question. If you care about understanding what is going on with the question that is asked, then wouldn't another example help you understand? Do teachers give one example per subject? And, the question I linked to came to mind as soon as I read this question, seeing as how I was a participant in this math contest. – Graphth Oct 30 '11 at 19:30
I think giving a similar example is actually useful. Thats why I upvoted this answer and also Steven Stadnicki's one. – Michalis Jan 23 '12 at 20:51

Here is an explanation I came across today.

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1
Note that this refers to the variant that doesn't have 0% as one of the answers. This is treated in the second part of Henning's answer . The present question, with 0% as one of the answers, not only leads to an infinite loop when you start out with 25% or 50%, but also does so if you start out at 0%; in contrast to the variant, there's no fixed point / self-consistent answer to this one. – joriki Nov 17 '11 at 12:48

The correct answer is c. 0%.

Either the conditions we are given are consistent or they are not.

If they are consistent, the way mathematicians solve these kinds of paradoxes is to say that none is provably correct in our formal system, however humans can use higher order reasoning to conclude therefore that c is correct.

If the conditions given are not consistent, then 1=2 and True=False, so c. is correct also in this case.

Conclusion: Only c. is always a correct answer

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9
Hmm -- this seems like an interesting answer at first, but in fact it's inconsistent :-) In the first part you rightly make a distinction between provable and correct, and then in the second part you suddenly drop that distinction and say that c. is correct just because we can prove it in an inconsistent system. – joriki Oct 28 '11 at 21:28

The question starts:

"If you choose an answer to this question at random, "

However it does not then continue:

"what is the probability that the answer chosen will be the probability of choosing that answer?"

It instead says:

"what is the probability that you will be correct?"

And then dosn't define correct.

There are 7 possible answers to the question:

a, b, c, d, 25%, 50% and 0%

However, lets presume that one of the following answers can be chosen:

a, b, c or d

The probability of choosing each answer:

a - 25%
b - 25%
c - 25%
d - 25%
25% - 50%
50% - 25%
0% - 25%

Being correct for "what is the probability that you will be correct?" if there is one correct answer (although as covered above the question doesn't define correct or specify how many answers are correct). This produces the same answer as the question "is the answer you choose the probability of choosing that answer?":

a - yes
b - no
c - no
d - yes
25% - yes
50% - no
0% - no

Now to what I think could be the real question in the question. "what is the probability that the answer chosen will be the probability of choosing that answer?":

For me the answer to this must be 'b' (50%).

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It is a trick question. let me rephrase the question to keep the misleading elements out

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

1. A
2. B
3. C
4. D

Provided that one of the four option is correct (assumption), the probability will be 25% that you are correct. Since you are picking up a random answer (don't bother about logic at all), the correct answer to the question "what is the probability that you will be correct" is 25%. Do not bother about the options, which is misleading.

Edit (Redoing and Correcting): I noticed there are two choices of 25% which is technically wrong because every choice must be different from the another. Lets redo the same example again.

Chances of each selection at random are 1/4 or 25%. That means 25% would select choice A at random, 25% choice B at random and so on. Since we know choice A and D are the correct answer (they are repeated which is wrong), that leads us to 50% correct answer if people make random choices.

So what is the probability that you will be correct is 50% :)

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6
-1 Sorry, if you're going to say "keep the thinking element out" and "don't bother about logic at all", and give a wrong answer (or at least change the question so the answer is different), I will have to downvote that. – Graphth Oct 28 '11 at 20:27
To make things complicated, it is like asking two questions in one . Best way is to simplify it. Otherwise it is totally misleading and not worth spending time. It like those optical illusions where a shape neither like cube, or box or ball or any regular shape out there. It is just an illusion. – TomCat Oct 28 '11 at 20:31

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That's the point. – Graphth Oct 28 '11 at 21:04

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@TomCat: You claim that "25% is the correct answer" and "the probability that you will be correct is 50%", which are contradictory. That's the whole point of the question... – Zev Chonoles♦ Oct 30 '11 at 20:10

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@TomCat: Read Henning's and Steven's answers above. Your argument is invalid because the question is "what is the probability that you will be correct?" and you are claiming both that this probability equals 25% and equals 50%, which is obviously nonsensical. – Zev Chonoles♦ Oct 31 '11 at 5:41
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## protected by Zev Chonoles♦Oct 28 '11 at 20:26

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