Wow, my little quiz received a lot more attention than I thought it would. Surprisingly, of all the answers that I have read so far (over 150 between the comments on my blog and Google+), I only spotted three that match exactly my reasoning, which in itself, has to be a statistical anomaly.

Anyway, enough waiting, here is my interpretation of the puzzler.

This is a meta quiz: a question about a question. The first step to figure it out is to solve the “inner” question, which is determining what “right answer” or means. In other words, we need to answer “If you have a choice between four options and you pick randomly, what are the odds you’ll get the right answer?”.

This is a very general question that can be answered easily: 1 in 4. 25%.

Armed with this knowledge, we now focus on the “outer” question, which we can rephrase as follows:

If you choose an answer to this question at random, what is the chance you will pick 25%?

Looking at the choices, we notice that 25% appears twice among four choices, so you will be right 50% of the time.

Therefore, the answer is 50%, right?

Wrong!

It’s B! You’re asked to pick a choice between A, B, C and D, not a percentage. Some teachers will actually fail you if your answer doesn’t typecheck ðŸ™‚

I think this is the answer that the creator of this quiz expected, however, the more I thought about it, the more I started to realize that there was a crack in the reasoning. Can you spot it?

It’s the answer to the question above:

If you have a choice between four options and you pick randomly, what are the odds you’ll get the right answer?

The answer will be 25% only if the answer is present **exactly once** among the options. Not zero, not two, three or four.

Do you think the meta quiz obeys this constraint? After all, 25% appears twice in the options, right? Yes, except that the correct answer is not 25% but 50%, which appears exactly once, so the quiz seems to be consistent with this caveat.

Exercise left to the reader: can you rephrase the original quiz to close this loophole?

#1 by

Christian Schlichtherleon October 29, 2011 - 8:59 pmWell, you’ve figured the difference between identity and equivalence. Answers A and D are not the same, but equivalent. In human languages, there is often no distinction made between the two – other than in software engineering.

Because answering this question requires some human intelligence, not artificial intelligence, I’ld vote for applying equivalence, not identity. Following this preference then, my conclusion is what I’ve explained on Google+, namely that this riddle is unsolvable.

#2 by

Christian Schlichtherleon October 29, 2011 - 9:05 pmFor completeness: You say: “If you have a choice between four options and you pick randomly, what are the odds you’ll get the right answer?”. The answer would be 25% if and only if the right answer is present exactly once. But this is not the case here. The right answer is present twice.

#3 by

Henry Buckleyon October 29, 2011 - 11:38 pmCan you really trust a conclusion which contradicts one of the intermediate steps used to derive it – i.e. an intermediate result that the correct answer is 25% is used to infer that the correct answer is B) 50%?

Is seems to me the difficulty here is one of language – I’m not sure the question really be parsed unambiguously.

#4 by

Andreaon October 30, 2011 - 12:08 amThis question isn’t simply a question about another question (inner and outer), but a question about the question itself: there is a recursion.

This bring my memory to the incompleteness theorem of göedel stating that a sufficient complex recursive axiomatic system has statement that cannot be decided: you cannot demostrate they are true nor false.

I think that the only way to make this question answerable is to eliminate the recursion and to rephrase it as a question about another question. (not the same one).

Andrea

#5 by

Peter Backlundon October 30, 2011 - 12:39 am“If you choose an answer to this question at random, what answer (option) corresponds to the probability that the answer (option) you chose corresponds to the probability that you chose the correct answer?”

Probability that you chose the correct answer: 1/4

Probability that you chose an option that corresponds to 1/4: 2/4 (A or C) = 1/2

Option that corresponds to the probability 1/2: B

#6 by

Fabio Mancinellion October 30, 2011 - 1:12 amI also agree with Christian. I think it’s unsolvable because of the fact that 25% is present twice in the list.

So if you focus on this then 50% is the right answer… But 50% is only present once which leads to a correct answer of 25% which is present twice etc…

This reminds me a bit the “Liar paradox” where you have to say if the statement “this statement is false” is actually true or false.

#7 by

Daniel Ribeiroon October 30, 2011 - 7:32 amI agree with Fabio. Imagine all answers were 97%. What would be the percentage? 0%. Is there an option that corresponds to 0%? No, therefore, there is no correct choice.

You can play self-consistent tricks like this, though:

If only one of them were 25%, and the others were != 25%, it would be 25% (only one choice is correct, pick that one).

If two of them were were 50%, and the others were != 50%, it would be 50% (any 2 of the choices are correct, pick any of them)

If three of them were were 75%, and the other were != 75%, it would be 75% (any 3 of the choices are correct, pick any of them)

If all of them were 100%, it would be 100%. All choices are correct, pick any of them

#8 by

Weiqi Gaoon October 30, 2011 - 7:59 amAnswer:

The answer is 0%. None of A, B, C, or D is the correct answer.

Proof:

The claim that C is incorrect is obvious: the chance can only be 0%, 25%, 50%, 75%, or 100% depending on whether 0, 1, 2, 3, or 4 of the four choices are correct. It can never be 60%.

The claim that A is incorrect can be “proved” by reductio ad absurdum. Suppose to the contrary to the claim that A is correct. Then D is also correct. Hence there are 2 correct choices, making the probability of choosing a correct answer 50%. This contradicts the assumption that A is correct. Hence A is incorrect.

The claim that B is incorrect can be “proved” the same way. Suppose to the contrary to the claim that B is correct. Then there is 1 correct choice, making the probability of choosing a correct answer 25%. This contradicts the assumption that B is correct. Hence B is incorrect.

QED

#9 by

Shiggity Shwaon October 30, 2011 - 9:08 amIf you add a fifth option, “E.) 20%,” that would satisfy the condition in the question. As it is written, none of the options satisfy the condition

#10 by

Timon October 30, 2011 - 10:07 amActually in your interpretation you don’t even have sufficient information to answer the question because you don’t know the probability distribution of the answers.

#11 by

Ryan Wardon October 31, 2011 - 3:58 amOk, I’ll give it a shot:

It’s an infinite loop. The 25% obvious answer is given twice which goes to –>50% chance (A & D) which means the answer is B (50%) and of course if you answer B then you only have a 25% chance of randomly selecting it which brings you back to 25% all over again. It also could be described as ‘undefined’ or a paradox. That is unless you change the rules by saying that there could be other answers not shown (like 0%). The question is intentionally vague. If one interprets the question to mean that you have to select one of the available choices, it can’t be solved. It is possible to solve it by giving an answer that assigns different weight to each of the 3 available choices or even by providing an answer that isn’t given, but if you stay within the limit of the question (one may presume that you must) there is no answer. I don’t think in that context it is 0, rather, it is undefined.

Ok, that’s my answer. I’m sticking with it –> Undefined

To answer the question differently would require that you add your own information to create an answer of your choosing.

#12 by

Sony Mathewon October 31, 2011 - 9:03 amThe answer is 0. Given that 25% occurs twice in the set, you must eliminate one immediately to still maintain a set of unique answers before you can calculate, in which case you get 1/3 = 33% which does not occur in the set and therefore the answer to the question: what is the chance you’ll get the right anwer is 0.

#13 by

hwiecherson October 31, 2011 - 10:04 amThis is how I see it.

The question is asking for a fixed point to the mapping for answers to probabilities of selecting them.

Basically, the relation that must be satisfied is

Probability(Choosing x) = x

and x can be 25%, 50% and 60%.

Because no choice satisfies this relation { i.e.

Probability(Choosing 25%) = 50% and

Probability(Choosing 50%) = 25% and

Probability(Choosing 60%) = 25% }, this question has no answer.

#14 by

Alanon October 31, 2011 - 1:00 pm1 – Answer B is correct because all the answers has a 50% chance of being right and 50% chance of being wrong.

2 – Exactly one answer excludes A and D that have the same values. So, between B and C (2 answers), only B is correct (50%).

#15 by

Jameson November 7, 2011 - 8:42 am“If you choose an answer to this question what are the chances of you being right” implies that it want the chance (probability) you are correct, since (B) is a statement not a probability it fails the type check argument.

Swap the inner question with “which is a primary colour (a) red (b) pink (c) green (d) purple” and now replying (b) makes no sense. The actual answers to the inner question don’t matter, all that matters is picking one and being right.

#16 by

Blackon November 12, 2011 - 2:44 amWe only have three answers to the test!!!!

So the chance to choose the right 33.33333 ….

#17 by

Blackon November 12, 2011 - 2:53 amWe only have three answers to the test if:

a) red

b) blue

c) green

d) red

Chance 33,3333333….. %

But if :

a) 25 type string

b) 50 type integer

c) 60 type integer

d) 25 type integer

Chance choose random right 25%

#18 by

Clayon November 17, 2011 - 8:07 amI saw this elsewhere and solved it. The answer is 0% as someone said earlier. That is the only stable value that completely satisfies the question. That isn’t a trick answer either, it’s completely correct. If the 60% option was changed to 0%, then there would be no correct answer.

#19 by

milashinion November 18, 2011 - 2:22 amthe flaw in your reasoning is that is circular! here it is:

assume that the probability of randomly picking the correct answer is 25%

then the correct answer is B

thus the probability of randomly picking the correct answer is 25%

#20 by

Jounion November 24, 2011 - 2:56 amThe right answer is as follows:

If we know nothing about the distribution, the chance to answer right by random can be anywhere between 25% and 50%. If we can assume that the distribution is uniform, the chance to get the answer right by random is 1/3. Thus, none of the given options is the right answer.

#21 by

leeon November 28, 2011 - 1:17 pmThis reminds me of the question:

How do you spell the english word pronounced the same as the Spanish word “tu”?

Or the alternate that you can’t write, only speak. Just read it aloud to see the problem:

“There are three ways to spell to.”

#22 by

David Hon November 30, 2011 - 12:36 pmWhat everyone is missing by focusing on “chance” and the percentages and the probabilities is the ludicrousness of the proposition being disguised by mis-direction, just as a stage magician distracts the audience from noticing a subtle movement needed to perform a “trick”.

The first trick is that THERE IS NO QUESTION!

People are first of all being sucked in by the declaration that “this” is a “question” and that it has an “answer” and that the answer will be “correct”.

Ask a legitimate question first. But there IS NO QUESTION to begin with!

The percentages are bogus distractions to make it “simulate” a question — waddling like a duck, quacking like a duck, ergo, it must also make a nice duck pate… but no question is ever posed that could even have a “correct” answer for which the percentages as false “choices” would apply.

Look again. Break it down semantically. It never actually ASKS a Question! It begs you to wander down a path focusing on CHANCE and you ruminate over the percentages as A, B, C & D but this example never gets around to asking one question for which there could be a “correct” answer.

#23 by

Mårtenon December 21, 2011 - 1:04 amThere is no correct answer free of contradictions.

The chance of being correct depends on the number of alternatives and how many of them are correct. In this case there are four alternatives. If 0 is correct the probability of being right is 0 %. For 1, 2, 3 and 4 correct alternatives, the chance of being right is 25 %, 50 %, 75 % and 100 % respectively. Thus the right answer has to be one of these five.

0 %, 75 % and 100 % isn’t available as an option.

25 % can’t be right since it’s available twice.

50 % can’t be right since it’s only available once.

For more headache consider the following example:

a) 25 %

b) 50 %

c) 50 %

d) 75%

If we say 25 % is correct, a is correct. The same is true for 50 % and b, c. Thus, In a sense a, b and c are all correct which implies the probability is 75 % which implies only d is right, but then the probability is 25 %…

In the following case a would be correct though:

a) 25 %

b) 33 %

c) 52 %

d) 90 %

#24 by

Gonon April 19, 2012 - 8:10 pmI assume that a, b, c, and d are possible answers to the given question. The universe here is constituted by the following parallel universes:

1) One of the answers is correct

2) Two answers are correct

3) The correct answer is not among the given possibilities

(there cannot be 3 or more answers correct, since the probability is only one, and the only repeated answer is 25%, being repeated twice)

Now, I consider the aforementioned scenarios:

CASE 1: One of the answers given (either a, b, c, or d) is correct.

In this case, without looking the actual answers, it’s easy to understand that the chance to be correct is 25%.

Looking at the four possibilities, we see that both a and d indicate 25%: there are two correct answers, which is INCOMPATIBLE WITH THE SCENARIO.

CASE 2: Two of the answers given are correct at the same time.

The only way for this to be possible is that both a and d would be correct, and then the actual answer for the question would be 50%. NICE. Now when we want to map the 50% to its corresponding letter, we discover that only one option shows that number, which is INCOMPATIBLE WITH THE SCENARIO.

We are left with scenario 3: The correct answer is NOT LISTED, hence the answer to the question is 0%.

In this “answer” post, you are actually CHANGING THE QUESTION. However, I guess that is what some teachers would do, but does not justify B being correct whatsoever!!