## Statistics Question

By Shamus
on Oct 30, 2011
Filed under:
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Here is a fun little question I found, via Flowing Data.

Now, this is very clever, although I really wish the answer for C) was 0% instead of 60%. See, if that was the case then it would make a nice circular puzzle:

1. The chance of randomly choosing the correct answer out of four is 25%.
2. Clearly the correct answer is 25%.
3. But, 25% appears twice in this list meaning the odds to randomly choose the correct answer of 25% is actually… 50%?
4. Okay, so 50% is the answer. But I only have a 25% chance of selecting that one.
5. I see. So this is actually a paradox, and the odds of selecting the correct answer is really 0%.
6. But… 0% is one of the options in the list. Which means I have a 25% chance of choosing it!
7. GOTO 2

I’m not sure what the 60% is. Maybe it’s just a random wrong answer, but I can’t escape the notion that maybe 60% has some meaning that’s going over my head. Is there some application of statistics where (through averaging or some such shenanigans) you could end up with an answer of 60%?

In any case, I got a laugh out of it.

EDIT: Comments were inexplicably disabled for this post. This has been remedied.

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1. Daemian Lucifer says:

“EDIT: Comments were inexplicably disabled for this post. This has been remedied.”

I was wondering about that.But then again,statistic flamewars can be pretty nasty.

I share the sentiment that the puzzle would be better with the 0% option.Still its a nice brain teaser.

• ClearWater says:

I read the question then nature called. While I was pondering the question I thought: answer C should have been 0% to make it more interesting. Then I came back and read the rest of the post. :)

2. The 60% is the freeby wrong answer. It is also a spillover check. 25% is the jump to answer without thinking enough, 50% the correct answer and 60% the obviously wrong check to see how many people are doing only at the error rate for complete misunderstanding.

3. TheMerricat says:

The actual answer to the question is 0%. The question doesn’t ask you to pick one of the values listed, it asks you what the chances are that you would pick the correct value from the list provided, if you picked at random.

• Ragnar says:

The question doesn’t even mention the provided list. The provided list could be completely unrelated.

Also it doesn’t give any clue as to what constitutes a correct value.

• decius says:

Suppose the domain of the answers from which you are randomly choosing is an infinite set (for example, numbers between 0 and 1 inclusive), and that there is exactly one correct answer. Then the answer is 0. This also holds if there are a finite number of correct answers.

If we instead assume that the random selection is from a finite set, then the odds of correctly guessing the answer is equal to the number of correct answers divided by the number of possible answers. Clearly, if 25% is the only correct answer, then 50% is the correct answer; that is impossible. If either 50% xor 60% is the correct answer, then 25% is the correct answer- which is impossible. If 25% and 50% or 60% are correct, then 75% would be correct, which is not listed, and therefore wrong.
1001=0100-> impossible
1000/0001/1100/1010/0101/0011->impossible because (a) implies (d)
0010=1001->impossible
therefore 0110 (b and c) are correct answers. The odds of selecting one of those at random from the set of four is 50% or 60% as we have just shown.

• Ragnar says:

Unless the probability distribution of your random number generator has a really high peak around the 0.

• decius says:

I’m not using a random number generator. I’m discussing a random selection, not a deterministic one.

• Ragnar says:

The RNG was a sidetrack. What I mean is the probability distribution. Just because you have 4 choices doesn’t mean that each choice is equally likely. Think of a weighted dice.

• John A says:

At first I thought the answer is “0%” for a similar reason. If we assume some correctness requires infinite precision.

However, allowing an answer “assuming X, the answer is Y” means we have to include this in the sampling distribution. Of course X needs to include all possible assumptions, even incorrect ones…

• decius says:

If the answer is in the form of the multiple choice options given, then B and C are correct. If the answer is in a free response form with no limits on the acceptable responses, then the answer is 0. If the answer is in a free response form with some limit (e.g. must fit on a sheet of paper, or limited to 255 characters), then the answer is sigma>0, since there are finite different responses.

• Halceon says:

How are the odds of selecting one of two answers from a set of four 60%?

• decius says:

As shown above.

• Syal says:

Everyone knows C is more likely to be chosen anyway.
At least I think that’s what he’s getting at.

It’s a multiple choice question, so those are the only possible answers. You’re just trying to find a way to get around it :D

• Ragnar says:

Of course. The fun in answering test questions is how to justify answering “The question does not make sense” or “The question is underspecified”.

• Nick says:

There were so many questions in tests when I was in high school and college that were like this. At first glance they made sense, but after further examination, they were incomplete and could not be answered. After a while I got tired of the examiner not knowing the correct question and saying “answer it how you think it should be answered” and just presumed the initial meaning was the correct one.

• Tizzy says:

Basically: we’re testing you on whether you learned what the questions should be and how we like them answered. Artificial knowledge rather than useful skills. Sigh!…

• Yes. That is an important skill for taking exams from some sorts of professors. I hated that sort of thing in school, but it is fun to look at them when the answers don’t matter.

4. Brandon says:

I am just amused because when I posted this, the comment count said this: “Three comments. 33% of them are the most recent.”

Seemed appropriate. :p

5. Mark says:

I believe it’s also making reference to the supposition that multiple-choice answers are not evenly distributed, which, in some cases, is true.

6. TheMerricat says:

A long time ago in a far far away land, before the internet, a teacher in a high school computer class played a trick on his students and told them that the computer he had setup in one corner of the room had been dialed into a powerful AI the local university had been working on and allowed them to type questions into the computer for the AI to answer.

Most of the students chose to ask mundane, chatbotish questions.

However, one of the students, me in fact, chose to test the AI’s problem solving skills and decided to pose a question familiar to many RPG and fantasy movie fans.

Sadly, having at this point limited typing skills, and being rushed on time due to the teacher being savy to the student’s personality and placing them at the end of the line, the problem was mis-stated.

The original problem, known to many, is:
“Before you are two doors, one which leads to riches and safety, the other which leads too pain and death. Next to the two doors are two guards. One guard always lies, the other always tells the truth. You may ask only one question and open one door. What do you do?”

The actual problem as stated was:
“Before you are two doors, one which leads to riches and safety, the other which leads too pain and death. Next to the two doors are two guards. One guard always lies, the other always tells the truth. What do you do?”

The response was “I open both doors and go through the one that leads to safety and riches”.

At this point the student realized he was being used as part of a Turing test.

• lazlo says:

Always remember you can get extra credit on a Turing test by convincing the tester that he is a computer.

7. Some Jackass says:

Wouldnt the answer be A or D? Since its asking what the odds are, which’ll always be 25%?
I dunno, my Stats teacher was an idiot.

I wouldve written E) 0% and circled that

• Moriarty says:

but if you would add an e) 0%, the chance of picking the right answer wouldn’t be 0% anymore, it would be 20%, making e) false again.

• Bodyless says:

No it would not make it false since you cannot guess an answer which is not provided.
Answer e) only exists because you create it and you cannot “guess” at the same time unless you put in some completely random number. But then the probabability of guessing the right one is zero in theory, since there is an infinite amount of numbers.

• Moriarty says:

No it would not make it false since you cannot guess an answer which is not provided.

that’s exactly the reason why 0% can’t be in the choices to pick from.

it isn’t possible to randomly pick, given a-d, so it is both the answer to the question and the probability of the correct answer to be picked.

if you discard a-d completely, it is still the correct answer because it is impossible to randomly pick the right number from infinite possibilities.

• Some Jackass says:

But if E isnt the right answer because it wasnt selectable, then there stands a 0% chance of the answer being right…but E is 0%…

• Well, no.
Consider if a, b, c and d all said 100%. By some yardsticks, that would make the answer 100% accurate and so you’d be right if you answered any of them.
The 25% (twice) and 50% (once) actually creates a weird fugue. I mean, if you’re taking these percentages to be relevant to answering the question. Because, if 25% is true, and there are 2 answers of 25%, then you can get the right answer twice out of four. If you can get the right answer twice out of four, the right answer is 50%.
But if the right answer is 50%, then 25% is the wrong answer, and so the right answer does not appear twice. If the right answer is 50%, it only appears once, which is to say 25% of the time. But then the right answer isn’t 50%, it’s 25%.
Choosing either of those answers to be true immediately makes it false.

But as other people have pointed out, it’s not clear what the question means, or even can mean. It seems to be semantically circular. What is “this question”? It’s self-referential; I don’t think it can have an answer because there isn’t a genuine question.
On the other hand, technically it doesn’t ask “if you choose an answer at random from the list below” what is the chance you will be correct, just if you choose an answer at random, period. By some readings it’s asking you to pick from the list a choice showing the chance you’d be right if you picked at random in general. In this reading it’s not completely circular, but the answer isn’t on the list.

So I see three possibilities. Taken the simple basic four-answers-normally-implies-25% way, all answers available are false in an interesting oscillation (except 60% which is just false). And yet the answer isn’t 0%, or at least it would stop being 0% if that answer was available, because if it was correct but it was one of the choices then the percentage would become (whatever was the % chance you’d pick zero)–another fugue.

Taken as a completely meaningless circular “question”, there simply is no possible answer.
Taken the third way, the answer is zero, but you’re not allowed to say so. I suppose you could imagine the list as a meaningless thing that just happens to be sitting near the question, in which case you could actually answer zero and get it “right” in some sense.

8. Jarenth says:

See, I tried to apply high school logic here, which led me to the following:

– A and D are both the same answer, so those can’t be right. I mean, that’d just be weird. Nobody ever puts the correct answer in a multiple choice question twice.
– That means only B and C are in the running.
– The chance of selecting the correct one out of two answers is 50%.
– Hey, that’s B!
– Man, I wonder what’s on TV right now.

• MelTorefas says:

I think this is brilliant. So elegant. Well done, sir!

• Mari says:

I did something similar. In 7th grade I had a class called “Study Skills.” Among other things we learned in that course, we were taught HOW to take standardized tests. Most of it was common sense stuff like “In a true/false test, answer false to any statement which contains words such as ‘never’ and ‘always.'” Some of it, though, was statistics based. We were taught that on a multiple choice question you should begin by eliminating the patently wrong choices. Then, if you know the answer to the question, mark it at that time. If you’re still not sure, chose B or C (an “inside” answer) as statistically the correct choice was most often an inside answer. Applying that logic to this question I immediately eliminated B since the probability in 4 answers didn’t seem to possibly come out to 60%. Beyond that, I had no clue and thus chose C.

• Mari says:

Edit: Eliminate C and choose B is what I meant.

• ehlijen says:

You were taught by the school how to make tests conducted by the school utterly pointless?

• Thomas says:

According to the Welsh education system I can speak GCSE Welsh based on my being taught that in multiple choice if a and b are similar and b and c are similar and d is completely different, b is always the right answer

I was taught the same type of stuff in school. The standardized tests are created by the state and simply run by the school. The school receives funding and accolades based upon how well its students do on passing the state’s standardized tests. Thus the school makes sure everyone knows all the tricks to make passing the tests easy.

• Mari says:

My class was shortly prior to that particular fad starting. The “you must pass a standardized test to get a diploma” started my senior year, exempting all seniors, and it was two years later that the state started tying school funding to performance on those tests. But clearly educators were well aware of what was on the horizon.

• Mari says:

Honestly, I expect that one of the aims of the class was to teach us to scam our SAT/ACTs. In addition to coaching on how to pass standardized tests we did vocabulary stuff using the SAT vocab lists, word etymology, and such. We also learned to “take notes” which was basically just learning various forms of outlining. How to do an MLA bibliography. How to organize notes for research papers. It was kind of a “catch all” class for things that had probably been added to the curriculum but the core teachers said they didn’t have time to teach. I’ve seen other schools do similar things. The public school my kids were in added a second period for science to the 8th graders’ schedule this year because over the summer the curriculum changed for what they were supposed to learn in 7th grade science. So one science period was the new 8th grade science and the other was the stuff that they didn’t learn in 7th grade but is now on the curriculum as “they should have learned it” though nobody in the state board of education can explain how changing the curriculum would have made it magically appear in the kids’ heads over the summer. It’s all unbearably bureaucratic but schools are left trying to figure out how to creatively use all that red tape.

• Tizzy says:

Nowadays, many such tests are computer-based and you can make the order of answers entirely randomized. I hope this will cure people from teaching this kind of BS study skills…

• Zukhramm says:

“If you’re still not sure, chose B or C (an “inside” answer) as statistically the correct choice was most often an inside answer.”

But, if they were aware of that, why not stop making tests that way and just put the right answer in all places an equal number of times?

• Syal says:

I remember being told that a particular test had an equal number of correct A’s, B’s, C’s, and D’s, and if you didn’t know the answer you should look at the spread of your other answers.

So basically, it doesn’t matter what they do, people will cheese it.

• Hmmm . . . so if they had a true/false test in this course, and it asked
“In a true/false test, always answer false to any statement which contains words such as ‘never’ and ‘always.’” (T/F)

9. Pete says:

Okay, lets see… at first you pick at at random. Assuming the chance of your selection being correct is uniformly distributed, you have a 25 percent chance of being correct. That means choices A and D are correct, which means your chance is 50 percent which means only B is correct which means you only have 25 percent chance of success which means A and D are correct which means you have a 50 percent chance which means B is correct which means aAAARGH CIRCULAR LOGIC

^ me attempting to sound smart-ish

• CaroCogitatus says:

Illogical! Illogical! Norman, correlate!

• popovitz says:

This is exactly what I thought when I pondered on the question myself…

10. Ragnar says:

100%, because my internal random number generator is highly biased. :-)

• droid says:

Well since any answer is as correct as any other, the chances are very high that you will give a best possible answer. 100% and 0% are equally valid responses.

11. Moriarty says:

heh, 0% is the right answer because it isn’t in the list.

I actually like it this way more, if it would be included in the posiblities you’d just have a liars paradox again. (“This sentence is false”)

• Syal says:

There’s no period, so it’s not a sentence, so it’s not a paradox. :)

12. Hitch says:

Could the 60% derive from the dubious advice to always guess “C” when you don’t know the answer to a multiple choice question. (It should be noted that it is not advised to use this advice in the case of a true-false test.)

• Daemian Lucifer says:

I thought it was always b.

• Kacky Snorgle says:

I used to listen to a radio station whose morning show had a weekly(?) call-in trivia contest. The name of the contest was “The Answer’s Always C”, and it involved multiple-choice questions. The amusing part was that the answer *was*, in fact, always C–and a very large fraction of the callers never caught on to this, despite the fact that the hosts kept repeating the name of the contest at every opportunity….

13. Thomas says:

I really want to find a paradox that isn’t self-referential. Barring that, there must be a consistent way to answer them rather than “this question is nonsense”. Something quantum :D

It’d only be a paradox if 60% was 0% like you suggested though, as it is, maybe it’s only 60% so that the question is answerable.

If you make it so the answer has to exist and just occupies differents states which it keeps switching between and the probability of it being in a certain state is a certain amount (and it can be in more than one state at the same time) then you could have a 4/5 of it being a 25% option and 4/5 of it being the 50%option. 1/2×4/5+1/4×4/5= 60%

But then 60% is an answer so the probability of 60% is higher than thought! The probability of getting a correct answer is 60% +1/4x the chance that the state is 60%. Since 4/5+4/5 =8/5 the chance of 60% being the right answer is 8/5 so the actual chance of being correct is 3/5+8/5×1/4= 3/5 + 2/5 =1. Since the chance of selecting one = 0 It stays fixed at one.

So you’re certain to get the right answer. Pick anything you like and made up a load of BS probability to prove that you’re right :D

• ClearWater says:

I have written the answer on a piece of paper and placed it in a sealed envelope. As long as you don’t open the envelope my answer is correct, but as soon as you open it the Quantum Multiple-choice Answer Function (or Q-MAF) will collapse into a wrong answer.

• Paul Spooner says:

Paradox forms are always self-referential, since they rely on self refutation. A non-self-referential paradox would be like a non-round circle. Thus, your wish for a non-paradoxical paradox is, itself, a paradox (being self-referential and self-refuting). Congratulations! You’ve constructed a single sentence meta-paradox!

• Thomas says:

As cool as that is (and it’s very cool) in maths they haven’t yet managed to prove that only self-referential questions create paradoxes. I guess I should have realised that it was a really stupid thing to wish for :D

I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

14. rrgg says:

60% is the probability that you will successfully argue with the test taker enough that he gives you the point anyways.

15. LVC says:

25%, 50% and 60% and four options to select one of these randomly, A,B,C,D.

Assuming 25% is the correct answer then you have 50% chance to select it.
Assuming 50% is the correct answer then you have 25% chance to select it.
Assuming 60% is the correct answer then you have 25% chance to select it.

So either 25% or 50% is the right answer. We can’t combine these into 60% unless we assume there are two right answers. However this puts us in an infinite loop because if we assume 25% is the correct answer, then 50% is the right answer which implies that 25% right answer, ect.

So assuming two answers, 25% and 50%, are right. We have 75% chance to select either one. This answer is not present to our assumption that there are two right answers is wrong.

Now this means that this is not a statistics question but rather a computer science question / math question. Selecting the proper answer is impossible because the question is self referential.

• Thomas says:

Statistics is maths :D apart from maybe some of the grimier interpretation. I also do think that 0% is a perfectly correct answer because 25% isn’t correct and neither is 50%

• Thomas says:

Ah my mistake, I forgot to chain together the last two sentences, which is a pretty appalling mistake. Stats is still a type of maths so I guess this is a pure maths problem would be strictly accurate but that would be very naziish :D

16. droid says:

This seems more a psychological experiment than a legit statistics question. I bet you got this question from a pre-test used by Aperture Laboratories.

17. Even says:

Correct answer is to cross over all the given answers and call whoever asked the question out on his bullshit and write that the chance is pure zero because the given parameters and consequent calculations lead only to answers that make no sense unless you blatantly break out from the given “equation”.

Alternatively:

1.25%
2.33%
3.50%
4.25%

• Aldowyn says:

This was my thought, but it doesn’t quite work. A and D are still different answers, after all.

• Halceon says:

Exactly. Only this time C isn’t an unrelated answer, it’s an intuitive one, albeit one that shouldn’t appear at all.

18. Nathaniel says:

I think the 60% for C might be a reference to the fact that C is the most common right answer on multiple choice tests.

19. Mephane says:

• Yar Kramer says:

Fun fact: the third item in Portal 2’s paradox list, “Does the set of all sets contain itself?”, is not in fact a paradox. If a set is defined as “containing all sets,” then of course it contains itself. No reason why it can’t. Russel’s Paradox is actually “Does the set of all sets which do not contain themselves contain itself?”

• Thomas says:

Although it’s a tricky set to define. Because you’d have all other sets in existence + “the set of all sets” as currently stand. But then you’ve created a new set of all sets + set of all sets. So the set of all sets contains that one too. But then you’ve created a set containg all sets + the set of all those sets and the set it create + the set that contains all those and itself. And so you’ve got to add that to the set and so on.

• Halceon says:

No, that’s just you thinking with your linear brain. A set that contains itself isn’t an infinite number of sets – it’s a single set. So if only the sets a, b, c exist and you define a = {a,b,c} you have successfully defined a set of all sets. The fact that this set is also {{a,b,c},b,c} and {{{a,b,c},b,c},b,c} doesn’t invalidate the initial set a being a set of all sets.

• Loonyyy says:

I’d reference you to the Cantor Paradox: The set of all sets cannot contain it’s own cardinality. Doesn’t that prohibit a true superset from existing?

• Wtrmute says:

By definition of the containment operator, every set contains itself:

∀A·A⊂A

If you want to explicitly deny this way out, you have to talk about strict (or proper) containment. Any set has itself as a subset, but not as a strict subset.

• Kacky Snorgle says:

Throughout this subthread, “contains” means “contains as an element”, not “contains as a subset”….

• Thomas says:

I’m with loony here. You are defining a set of all sets, you’re defining 3 objects and a set which contains those 3 objects.

You’ve answered “no” to does the set of all sets contain itself, which we just agreed above, is the wrong answer. Once you define a set of all sets, even in a universe with 3 objects, the set of all sets (being a set) must contain itself

• Andrew D says:

In most cases, “Does the set of all sets contain itself?” does create a paradox, merely because the existance of a set of all sets creates a paradox in itself.

To get around this, they just instead decided to say that while an object may exist which contains all sets, it cannot be a set itself.

• Thomas says:

Or they decide that the whole of logic is broken and completely destroy proof by contradiction! Paraconsistent logic FTW! I really really want to study it but my Uni hasn’t offered up a chance yet :(

• Zukhramm says:

While other might talk about trigonometry or calculus to me there is only Good Maths and Evil Maths.

Good Maths include calculus, geometry, linear algebra, some differential equations.

Evil Maths include topology, number theory, set theory, and probability.

• Loonyyy says:

Come to the dark side, we have cookies. Plus, we get to pretend to be crazy excentrics. The fundamental shapes are a projective plane and a sphere! A sphere is a cube!
I like more pure maths, so I guess that makes me evil… Which means I can get started on the mad science bit right away!

• Demo says:

Or as I like to call them, filthy applied maths and proper maths respectively.

• Thomas says:

Too right!

• Zukhramm says:

What? There’s applications to all of them and you can use any of them without applying them too.

20. NihilCredo says:

It could help, in order to ‘get’ how the paradox works, to read the question as follows:

“Is any of the following answers equal to the chances of its value being picked at random?”

• Pickly says:

Was going to try and make a joke post about this, bit it does seem the closest to a legitimate way to try and “solve” the question. (Or as close as you can come to solving it.)

21. Scott (Duneyrr) says:

When I first saw this problem, answer C was 0 causing me much grief.

• MintSkittle says:

So the answer is 100%, and you circle them all.

• Loonyyy says:

In which case, all of the answers are wrong, and the answer reverts to zero, which has a 25% chance, and you go back around again!

• Scott (Duneyrr) says:

If you’re going to bend the rules, you just write in ‘E. 20%’ and choose that one!

22. Mom says:

The term “this question” is undefined.
No answer can be determined until the term “this question” is defined.

23. Alan says:

When I looked at it, if you discard the random element, I thought that it was 60%.

Ignoring the self referential elements of a,b,d, if you apply game theory to it, 60% is the highest, non-looping number in the sequence.

I don’t think that it is correct, just my interpretation of it.

24. Hehe! Amusing indeed! The chance of being correct is 50% and is always 50%.
The probability of being correct on the other hand is 25%.

People always confuse chance and probability.

Usually when people say that there is a xx% chance of something, what they actually mean is that there is a xx% probability of something.

Remember that even if there is a 99.99% probability that something can happen, there is always a 50% chance that it might (or might not) happen.

For example dropping a rock from your hand right now will cause it to fall due to gravity. One might think that thus it is 100% probable.
But it is theoretically possible for gravity disruption.
But it’s so rare that dropping a rock is 99.99……9% probable.

It’s like throwing dice, there is a certain probability that it will end up with a certain number up, but there is always a 50% chance that any number may end up showing on top.

So when people state that something is impossible or there is 0% chance or so on,
I always like to say that “No, it’s merely improbable. But there is always a 50% chance of anything happening!”

Maybe one can say that chance = chaos and that probability = degree of order?

• Amarsir says:

I don’t suppose you have a source to back up that definition?

• Jakob says:

Chance: If event x has a chance of y % of happening, it merely mean the event has the probability of y% of happening and that the event is beneficial (i.e. the chance of winning the lottery). Risk is similar, just that the event is unfavorable.

• Loonyyy says:

I think he’s making up definitions to suit his theory. I certainly have never heard chance and probability referred to in this way. The difference in usage is usually that Probability is also used to define the field of study.
With a dice, there is never a 50% chance of anything. If you define it as being right or wrong, you still will find that wrong contains a set of equal likelihood which is 5X that of you being right. You’ve always got a 1/6 chance of a stated exact outcome occuring, assuming of course, you aren’t hoping for a number not to come up. That’s a definable fact. I don’t know where you’re getting your definition from, but I would need a very convincing source to believe this. And it still doesn’t solve the problem.

• TSED says:

Time to be unnecessarily pedantic:

“Getting a number >10 on a d20.” There, that’s a 50% chance.

• Cragfire says:

I’ve never heard of the word chance being used this way. Chance is frequently used interchangeably to mean “probability” or “possibility.” Neither one of those resembles your strange usage. Possibility is a binary state. A thing is possible, or it is not. There are no percents involved.

• Zukhramm says:

Generally, chance is used when the outcome is considered positive. If it’s negative it’s a risk instead.

• I’m pretty sure this doesn’t work. The question is impossible to answer because there’s no 0% option so even by your (pretty certainly incorrect) definition of chance, it’s not 50%.

• decius says:

If something happens 99.99 times out of a hundred trials, there is a 50 out of a hundred trials chance of it happening.

You just unilaterally redefined ‘chance’ and/or ‘per cent’.

• Loonyyy says:

I agree. 50+99.9999999 sums to greater than 100%. That means that more outcomes can occur that will occur. So an impossible outcome is statistically likely. That doesn’t happen. Mutually exclusive outcomes always sum to 100%, that’s axiomatic.

EDIT: I found it: http://en.wikipedia.org/wiki/Bayesian_probability . He’s using Bayesian probability, which is not a mathematical definition of probability as pertaining to chance, but as pertaining to information and knowledge. Which would explain why he still didn’t answer the question.

• Zak McKracken says:

Wait a second, according to your interpretation of “chance”, the chance of winning the lottery is also 50%? Because I either win or I don’t, no? Two states, so 50% chance.
The reaaaally cool thing about this is that it’s independent of wether I actually took part in the lottery.

Or did I get that completely wrong?

• Tizzy says:

Here I am fervently praying that you have trolled us successfully… The possibility that you were serious is too frightful to consider.

25. Mike C says:

Perhaps it’s a psychological question. People, when given a random choice, still put thought into the answer (as everyone is doing above). They also want to be right, so they want to pick the answer that groups them with the majority, which is C) 60%.

Therefore, the 60% of people who think this way will choose C. The other 40% will logic themselves into A, B, or D.

26. Ramsus says:

False!

27. The answer is PONIES! Goodnight!

28. Matt says:

The question specifies “at random” but doesn’t specify even weightings. So if you weight the answers properly in your random selection, you can make sure C is correct.

29. Daemian Lucifer says:

Speaking of paradoxes(not really,but I needed a clever segue),Shamus have you tried out achron?

30. mac says:

@Roger Hågensen: You are probably the most hardcore bayesian I have ever encountered. Congratulations!

• Heh! I actually had to wiki Bayesian (even though I’ve heard the word before), and thanks I guess *laughs* (my ego demands I take that as a complement), and yes I’m heavily “logic” driven in my views of the universe.

Then again as I’ve stated several times in the Twenty Sided comments over the years I’m an Absurdist (look up Absurdism on wikipedia for a good description for example).

Now whether my own behavior is logical or not is a completely different matter. My views does not need to match my actions, although that would probably be preferable.

And I agree, the commenter on Twenty Sided is actually a lot smarter than, well most other places on the net.
Or at the very least there is a higher percentage of “thinkers” here.

Oh and the way the question is phrased on that chalkboard is very ambiguous.
I only answered with chance and probability as that was the subject of Shamus’ post.
But the actual question does not restrict the answer to the lettered alternatives, if it had said “Which of the following answers…” or similar.
The question does not prevent one from making your own “E” answer.

PS! You might like (unless you already are familiar with) the TV show QI http://en.wikipedia.org/wiki/QI
In fact I highly recommend anyone here to see that show if you like facts, truths and unknowns with a good splash of British humor.

I have almost no intuition so I am painfully aware of just about everything that I assume. I had to use this kind of logic all through out school to try to figure out what in the world that crazy teacher meant by that bizarre self contradicting question. Obviously this question isn’t intended to have a right answer, but that is what my brain kept wanting to do. We really can’t function without implying, presuming, and assuming, so next time someone says “you know what happens when you assume” punch them in the nose for me.

Nowhere is it stated that the arrangement of chalk on this board is intended to phrase a question! :P

• Mom says:

exactly

• tussock says:

Bingo.

You randomly choose, you get 25% a), 25% b), 25% c), 25% d). No matter which of those answers is correct, it must read 25% to be correct. That leaves a) and d) as possibly correct.

a) and d) cannot both be correct, as that is only true if it’s false. No combination of correct answers leaves us with any correct answers.

So there’s some chance a) is correct (P), and a related chance that d) is correct (1-P).

25% * P + 25% * (1-P)
=> 25% * (P+1-P)
=> 25%.

So that works. If either a) or d) are correct then either a) or d) are correct.

The fact the question gives us no way to find the correct answer by deliberate choice does not change what the correct answer is.

32. The question is meaningless. Are we assuming that you can only have four answers, that it’s a true multiple choice with no trick questions? If so, then it IS 25%. But what if “this question” could be ANY probability as an answer? Then it’s 1%. It’s a silly question…

33. Kacky Snorgle says:

Okay, so the answers are right because they’re wrong because they’re right. Yeah, whatever.

I’d be more impressed if the puzzle were both self-referential *and* consistent. Y’know, like this one. :)

• Paul Spooner says:

That looks like a pretty hard test. How would you go about solving it? It seems like a trial-error approach would be the best to start with; Maybe going from last to first?
Looks like crossing out wrong answers is working so far. It’s like a terrible twisted Sudoku puzzle.

• Kacky Snorgle says:

That’s pretty much the idea. Question #20 is one obvious place to start; you can also play off #10 and #16 against each other. After that, you quickly end up in Sudoku territory, crossing off impossibilities all over the place until correct answers become clear…. (I’m pretty sure I saw this test long before I saw a Sudoku, but I don’t know which is actually older.)

• Paul Spooner says:

Managed to solve it (finally) after several false starts. The fallacy of denying the antecedent caught me a few times before I realized I was doing it. Excellent logic puzzle! I much prefer it to the paradox kind.

If anything, I wish question #20 was also self-referential. Something like: “The answer to this question is (A)E (B)E (C)E (D)E (E)E”

34. Ellery says:

Am I the only person who actually chose an an answer at random, then read what my choice was?

As others have discussed in passing above, I just chose “C” because of the multiple choice rule “choose C if all other answers seem equal”. I hadn’t read any answers yet, so I chose “C”.

I can be quite literal.

• Indy says:

I think this “logic puzzle” is a view into psychology and 60% of people pick C as their random guess. That means I pick C.

I am probably wrong.

Also, 100% of the statistics in this post are wrong.

• Dev Null says:

The original question doesn’t tell you to pick an answer at random, it asks you to predict the results if you _did_ choose an answer at random.

35. HiEv says:

The difference between humans and computers is that computers would get stuck trying to figure it out because of answers A, B, and D. Humans, on the other hand, quickly get past the paradox and get stuck trying to figure out why C is 60%.

• Amusing tidbit:
When I ran the following code, it ended up with 60% as the answer on the first try (hows that for chance? O.o).
Obviously this is merely one of many ways to programaticaly write the um… “problem” shown on that chalkboard.

```;PureBasic source, can be run as is in the PureBasic demo at http://purebasic.com/ a=25 b=50 c=60 d=25```

``` ```

```Repeat answer=Random(100) ;0-100% If answer=a Or answer=b Or answer=c Or answer=D Break EndIf Forever Debug "Answer is: "+Str(answer)+"%"```

This code could be improved by actually gathering statistics while randomly picking numbers, and then calculating the actual percentage, but I’m also lazy by nature so…

36. DoctorSatan says:

60% is not possible i think because there are only 4 options.

heres what i thought of the question:

if this is a MCQ with only one right answer then the answer is b) 50% because then options a and d become invalid(because only one option can be correct). so only 2 options matter => 50%

if this is a MCQ where multiple answers can be correct then i would have left the question because screw maths!

also i think this is probability.

^^^^^ <—dont flame me. im not good at maths. again… screw maths!!!

37. Richertai says:

Someone posted this to Reddit. I spent about 5 minutes laughing about it and concluded that 0% should’ve been an answer. Nice find.

38. This is a response to pretty much everyone who disparage what I wrote above!

http://dictionary.reference.com/browse/chance
1.
the absence of any cause of events that can be predicted, understood, or controlled: often personified or treated as a positive agency: Chance governs all.
2.
luck or fortune: a game of chance.
3.
a possibility or probability of anything happening: a fifty-percent chance of success.

http://dictionary.reference.com/browse/probability
4.
Statistics.
a.
the relative possibility that an event will occur, as expressed by the ratio of the number of actual occurrences to the total number of possible occurrences.
b.
the relative frequency with which an event occurs or is likely to occur.

And before anyone complain about the picking of definition 4, please note the subject of Shamus’ post: Statistics Question

You can argue as much as you want with me about theories or philosophies, but facts or empirical data or dictionaries one can not argue with. (though I guess you could argue with those keeping the dictionaries, but that has nothing to do with me in this case.)

• Loonyyy says:

That still isn’t proving you right. In standard probability, chance is synonymous with probability. The two have different definitions, but not with regard to a stated outcome occuring given a set probability. And you’re still answering problems you created rather than the problem at hand.
Chance and probability do differ, sure. But you made many mistakes in your post, like assigning exclusive options which totalled more than 100%, and also, if you choose option 4 as your definition, then option 3 in the definition of chance is synonymous. A ratio can be converted into a fraction, and vice versa, which is why probabilities in gambling are given as odds.
And since you want to argue from authority: This online Thesaurus holds Probability and Chance as Synonymous for some purposes of definition (No, I don’t think providing a link to semantics proves me right, but if your argument is unassailable, then so is mine, Q.E.D.)

And stating that you can’t argue with empirical facts is correct. Stating that you can’t argue with a standard dictionary for common english usage is not. Definitions are changed and subjective. It’s not the same as evidence, especially since you didn’t seem to notice that your own source said the two were the same whilst you were cherry picking.

• Shamus says:

You are trying to “win” an argument on the internet, instead of having a discussion.

You’re being needlessly provocative and confrontational over what is a misunderstanding over the finer points of a trick question.

• Hehe, so true!

@Shamus I’m not sure whom you mean by “You” as this could indicate me or Loonyy, or both of us or all here.
If you are indicating me specifically then I apologize for not having the language skills necessary to express myself properly, I’m not a native English speaker, although that is no excuse either, I’ve always had issues turning thoughts and ideas into words that others understand.

@Loonyy Yes definitions can be changed and subjective, but as I stated that is an issue one might take up with the dictionary keepers. To me a respectable dictionary is empirical as I have no other vetted source of definition of a term or word.
Oh and please do not take the lack of any smileys in my postings as a sign that I’m angry or confrontational, there is never any malice behind my words, if there where then I would use actual words that would leave no room for any other interpretation at all, so if I offended you in any way then I must apologize again, not for what I wrote but that you where offended by it instead.

It’s surprising how hurt people can get over words, I fully separate words from actions, words ‘can not’ or (logical or) ‘should not’ hurt people, physical actions on the there hand can.

I have no idea what in my first post can be considered flamebait, nor was that the intent. When questioned I re-iterated or tried to clarify or even source references to help clarify, at least if I’m wrong other would know where (if any) errors originated from.

However I must commend “mac” further above, if I had known of the Bayesian probability then I would have stated at the start that under the Bayesian probability theorem I believe the following…”

• krellen says:

My brother uses that as his avatar online. That should tell you all you need to know about him.

• Daemian Lucifer says:

“You are trying to “win” an argument on the internet, instead of having a discussion. ”

Wait,you arent supposed to win?What a shock!

Ha!I was right.Statistics flamewars ARE nasty.I wins da interwebz!!

• PS! The only one I would ever consider an actual flamewar with would be Rutskarn, not out of malice but because it would be fun, I’m pretty sure we could easily match each other’s wits, after all, “originals” like us are rare, even on the ‘net. *laughs*

• Zukhramm says:

But you said “The chance of being correct is 50% and is always 50%.”. Which none of those definitions, nor any I have ever seen seems to imply.

By the third definition, which is the one we’re really talking about here, it’s the same thing as probability. The links you posted disagree with your definitions rather than help them.

• Kacky Snorgle says:

At the risk of becoming involved in a rare Twenty Sided flame war, I will attempt to comment constructively:

Your dictionary’s definition #4 of “probability” leaves out the key point that the outcomes under consideration need to be equally likely in order for this definition to apply. If you roll a fair d6, there’s a 1/6 chance of rolling a 3, by this definition. If the die isn’t known to be fair, then the chance of rolling a 3 might be anything between 0 and 1, and we can’t compute this chance just by knowing that the die has 6 sides, or just by knowing that we’re sure to roll either a 3 or a non-3. The best we can do is to experiment with the die for a while and try to work up a reasonable estimate, and that’s where Bayesian probability comes in.

I’m not sure whether the dictionary’s definition should be considered wrong, or merely nontechnical. At any rate, it isn’t the sort of careful definition you’d want to use if you were trying to do calculations. Indeed, a common error in probability calculations is to use this definition on outcomes that aren’t equally likely.

But I don’t see what any of that has to do with a distinction between “chance” and “probability”. Indeed, definition #3 of “chance” is “probability”, and that’s certainly the sense in which “chance” is usually used in discussions like the present one…. I don’t see any definition of “chance” on your list which would support your earlier assertions. I therefore remain confused by them.

• Shamus says:

For the record, I’ve heard other people use the definition that Roger Hågensen has used, and they offered it as something they were taught in a classroom. I was hoping one of those folks might chime in here, since this is something I’ve picked up on the internet. I’ve never had a statistics class myself.

The idea is that “chance” is used to describe various possible outcomes without regard to probability. As in, “There’s a chance that might happen.” (This unlikely event is still within the list of possible outcomes.) Using chance this way, you could list all of the possible outcomes of 3D6. In such a list, you would simply list all of the numbers from 3 to 18, even though 12 is far, far more likely than 3 or 18.

I don’t think arguing with a dictionary is a particularly useful line of discussion. Even if that definition WASN’T in the dictionary, if it’s a definition that people use then it might possibly have been used by the person proposing the original question.

I can see the utility of having a word for “things which are possible outcomes without regard to likelihood”. Seems like it would be a pretty important concept in discussing things like this. It’s unfortunate that the word “chance” is commonly used to specifically describe probabilities. This is a recipe for confusion and irritation. (I offer this thread as proof.)

• mac says:

possibilities?

• Kacky Snorgle says:

It’s just a matter of carefully defining your experiment. If the experiment is “Roll 3d6 and record the sum”, then the three dice are treated as indistinguishable, the individual result of each roll is ignored, and only the sum matters. There are 16 possible outcomes {3, 4, …, 18}. But since those outcomes aren’t all equally likely, the fact that there are 16 of them tells us nothing about *how* likely each of them is. So the number 16 might well be useful to us, but the number 1/16 is certainly not.

If we want to calculate the probability or chance of any particular sum, we have to go back and distinguish the dice — label them as die #1, die #2, and die #3, or some such thing. The experiment becomes “Roll 3d6 and record the sequence of rolls”, and the outcomes are now ordered triples like (4, 5, 1). There are 6*6*6=216 such outcomes. But unlike the 16 outcomes (sums) above, these 216 outcomes (triples) are all equally likely (under the assumption that the dice are fair). So unlike the number 16 above, this number 216 can meaningfully be used as a denominator. Then to compute the probability of any event (e.g. a particular sum), we just need to count how many outcomes (triples) belong to that event (have that sum), and divide by 216.

If Roger, and you, are merely saying that the word “chance” is sometimes used informally in contexts where such calculations aren’t intended, then you’re certainly correct. But it sounds to me like you’re suggesting that the word “chance” or “probability” could be applied to the number 1/16 in the above example — and that’s simply nonsense. That number doesn’t have a name, because that number shouldn’t come up at all in this situation. We’re free to be nontechnical and leave out numbers entirely, but we’re not free to use numbers in incorrect and misleading ways. This isn’t “a definition that people use”, unless those people are half-remembering a long-ago math class.

A probabilist working with 3d6 would always begin with the sequence-of-rolls approach, precisely because it produces equally likely outcomes. He would say that the 16 possible sums of 3d6 form a partition of the sample space: that is, exactly one of them must occur on any given roll. Is that the sort of word you’re looking for?

39. Kevin says:

The answer, of course, is 42.

• Daemian Lucifer says:

Does that mean that Shamus actually has posted the question of life,universe and everything?He is the final product of millions of years of computations!

• Hehe! Thanks, my hat is off for the HHGTTG reference there, nice to meet fellow Science Fiction fans. and your answers is also philosophically correct as far as the phrasing of the question on the chalkboard is concerned, oh and you didn’t need to put a smiley in there to make people understand that it’s tongue in cheek as well.

• Ah, so D then (using 1-numbered arrays anyway, otherwise the answer is IndexOutOfBoundsException)

• If you liked that one, check out Derren Brown’s “The System”, search for derren brown coin toss or derren brown coin trick on youtube.
Later in the program he reveals how he did it. (without spoiling anything, him tossing a coin that landed on the same side ten times in a row was not really a trick at all).
An awesome demonstration on how probability/chance/randomness works, it usually open the eyes of those who have misconceptions on stuff like this.

40. Abnaxis says:

Hm…

Y’know, just because you are picking answers at random, doesn’t mean every answer has the same chance. So all I need to do is write a RNG that gives A and D 12.5% of the time and B and C 37.5% of the time, and I would be right to answer A or D.

• Exactly. The question is one of those questions that, in a class, I would either answer “Not enough information to answer” or ask for quite extensive clarification.

41. If anyone is interested, this wiki article has a very nice uncertainty chart that lays out most of the various views/reasoning found in these comments:
http://en.wikipedia.org/wiki/Uncertainty

Interesting because the question on that chalkboard is very much uncertainty laden.

42. Zak McKracken says:

Completely different question, only slightly related to the picture posted:

Why do I always see “No water on this board!” signs next to blackboards (er… greenboards?) in US classrooms, in about every Hollywood movie? Is that a Hollywood thing, or is it actually not allowed to clean a blackboard with water in the US? Because, you know, it would not occur to me to use anything else … and using a dry sponge on chalk is sooo … *shivers*

• Abnaxis says:

Wow. I have to think back to elementary to even remember what they did to clean chalk boards (they mostly used dry-erase-marker boards at high school, though a few classes were stuck with chalk-and-slate). Now I feel old…

I seem to recall that, most of the time, we just used erasers. When the board got too messy from build up, the teacher had some sort of cleaning spray. I always presumed they used the spray because it was more effective than straight water (I thought it was just water with some soap mixed in) but maybe there was more to it than I knew. I never asked, and we didn’t have any signs posted–a lot of times, it was a miracle if the board ever got cleaned with anything :D

• asterismW says:

I’ve actually never seen one of those signs, either in movies or in real life, and I’ve seen my fair share of chalkboards. We always used water. Still do, in fact.

43. KremlinLaptop says:

Is this a thing? Is this actually a thing we’re doing now? …Seriously?

We’re really going to get the comments closed on a question about statistics (and it is going to happen, a few more snarky posts, some more “NO. You’re WRONG!” sorts and we’re there). Not the religion post from the autoblogography, but rather statistics is the Achilles heel to our so civil comments sections?

…Truly, we are nerds.

• Thomas says:

How dare you suggest that statistics isn’t important! … I actually don’t want to type a load of flame so just assume that I carried on and mentioned your mum a couple of times…

• Tizzy says:

I’m not going to say he’s wrong… I’m going to say he’s not even wrong

44. drlemaster says:

Assuming this was asked at my high school, the answer is 25%. Back when I was in high school (I’m about 3 years older than Shamus), all multiple choice tests where graded by a Scantron machine. The Scantron machine could only support one correct answer for each question, so the correct answer for this question would be 25%. Sadly for anyone taking the test, they would have no way of knowing whether the teacher had coded A or D as the correct answer. So even if you choose the correct answer, the probability of getting it correct is only 50%.

• asterismW says:

So true. I remember that happening. “Wait, that answer is listed twice. Do I choose A or C? I’ve got a 50% chance of getting WRONG the answer I KNOW is RIGHT! Aargh…”

45. SteveDJ says:

Read the question again. It concludes by asking “…what is the chance you will be correct?” This part of the question has nothing to do with the choices below it. So, the correct answer is not A, nor B, nor C, nor D. Those four choices are part of the QUESTION – thus, provided for the purposes of making a random choice.

Nobody said the question itself was a multiple choice question.

46. GSP says:

What does “correct answer” mean in this context?

It means “probability of randomly getting this answer is equal to numerical value of this answer”.

So the question can be rephrased as:
“What is the probability (1) of choosing an answer that has a probability (2) of being chosen equal to its numerical value?”

So there are actually different probabilities. They sound very much the same, but really, they are not.

(1), the one we are being asked about, is probability that we get an answer that meets our correctness criteria by answering randomly.
(2) are probabilities of each individual answer occuring.

Probability (2) – value pairs we have are following:
50% : “25%”
25% : “50%”
25% : “60%” (or “0%” – it changes nothing)
As you can see, none of the answers meet our correctness criteria (prob == value).

So the probability (1) of an answer being correct is 0.

People before me who said this isn’t a multiple choice question are right. The whole question logic is just very, very cleverly disguised to make it look like an infinite recursion loop.

• Oh, I like this one. It’s the reverse of some odd parsing I tried, and actually gives you a genuine answer that you are allowed to answer.
I was suggesting the question could be parsed as asking you to choose from the list an answer that would equal your chance of being correct if you answered randomly in general.
You on the other hand are construing it as asking in an open-ended fashion, what would be your chances of being correct *if* you randomly chose an answer from the list–and as you say, the chances of being correct if you did so are zero.
That’s great! The only downside is I’m not sure the grammar supports that interpretation.

47. Roy says:

I must not be as good at statistics as I thought, because I keep getting the answer as 1/3, which isn’t an option.

What are the odds that 25% is the answer? Assuming the answer is uniformally distributed between the three distinct answers (25%, 50%, or 60%), it’s 1/3. And the odds that I pick 25% out of the A-D choices is 2/4, making the odds that I pick 25% as the right answer being 1/6.

The other two? The odds that 50% is the right answer is 1/3 (again, assuming uniform distribution), the odds that I pick it are 1/4, so the probability of 50% being the selected correctly are 1/12. 60% has the same answer. So the odds of picking the correct answer are 1/6+1/12+1/12 = 1/3. Which isn’t available.

That’s assuming the answer is uniform among the three – 25% is just as likely the right answer as 50%. If the answer is more likely 25%, due to it appearance twice, the odds change to 2/4 * 2/4 + 1/4 * 1/4 + 1/4 * 1/4 = 1/2. Which does appear.

So the correct answer is 50%, assuming that the answer is randomly selected between the four choices, not the three answers.

• Soylent Dave says:

But if 50% is the correct answer, then there’s only a 25% chance of you selecting it as such – which means it’s the wrong answer.

48. Soylent Dave says:

I’ve seen more than one study where a human interrogator would assign the correct answer (in a 4-answer multiple choice test of significant length) to ‘C’ nearly 60% of the time (with a 5-answer test it’s B most of the time, but I forget the percentage).

I was always under the impression these were relatively well-known studies (at least amongst people who like statistics) – it’s an example of humans being rubbish at doing random, and the patterns we create when we’re trying to avoid creating obvious patterns.