Happy Halloween! Last week I described how @jbrownridge had sent me this lovely puzzle….
If you choose an answer to this question at random, what is the chance that you will be correct?
a) 25%
b) 50%
c) 60%
d) 25%
If you have not tried to solve it, have a go now. For everyone else, the answer (kinda) is after the break.
Well, its’ a tricky one. Essentially, there are three answers – 25%, 50% and 60%. So, if you choose one of these at random then you will have a 33% chance of being correct. However, this is not one of the answers, so you are in a somewhat difficult situation. My guess is that there isn’t an answer, but I can see how different linguistic assumptions make for different answers. So….what did you think?
I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.
October 31, 2011 at 7:09 am |
Oh…Does this mark the first time that the answer actually *wasn’t* after the break?
October 31, 2011 at 7:10 am |
Yeah, next time can we go back to puzzles that have an answer please?
October 31, 2011 at 7:15 am |
WTF?
October 31, 2011 at 7:18 am |
I disagree. There are four answers, a, b, c and d. The fact that the text in the alternatives happen to be percentages is largely irrelevant, and the chances are still 25%
October 31, 2011 at 8:14 am
so does that mean u choose A or D then?
October 31, 2011 at 8:18 am
No, the choice is random, and the contents of the options is irrelevant. a) and d) say 25% but as far as this question is concerned, they could have said “sausage”
October 31, 2011 at 10:05 am
No. None of the answers can be correct because 25% and 50% contain contradictions. Therefore the answer is 0%, the chance the you will select the correct answer is zero. Imagine if the question was What is 2+2 and the answers were 1,3,6 and 2. None of those answers is correct but by your reasoning there is still a 25% chance of selecting the correct answer, there is a 25% chance that you will select any individual answer but if none of them are correct then you cannot select a correct answer.
October 31, 2011 at 10:18 am
Tort Says:
> No. None of the answers can be correct because 25% and 50% > contain contradictions.
What makes you think they are contradictions? What makes you think they are probabilities, or even the pool of possible answers? Which part of the question told you anything at all about the nature of the lines that started with letters?
You are right that 25% is wrong, I already changed that in a later post. The correct answer is not 0%, but asymptotically close to it
October 31, 2011 at 2:33 pm
The answer is still 0%, as there is no correct answer given. The question asks about probability, which is why the fact that the answers are given in percentages implies that there is a correct answer. If 25% were given once, then that would be the correct answer, but since it is given twice, you have twice the opportunity to choose it, which makes the answer 50%, but then if the answer is 50%, you can only choose that one way, making it 25% again.
November 4, 2011 at 3:27 pm
Well, to me the answer is 50%. The question is not “Which answer is right, a-b-c- or d?” The question is, “What is the chance that you will be correct?” So, usually, if you would answer randomly, you’d get 25% of chance to be right… but since there are two choices that says 25% (they could have said sausage like someone else said, the values in the choices are irrelevant, they are there to confuse us), therefore, the chance to be right is now 50%.
October 31, 2011 at 7:19 am |
Multiple Choice questions are usually evaluated by a computer, in which case answer a) and d) are not the same. Chance you take the right one is therefore 25%.
October 31, 2011 at 7:23 am |
Actually, you’ve got a 50% chance of picking “25%”, a 25% chance of picking “50%”, and a 25% chance of picking “60%”.
So presumably the answer is “0%”.
(it would be more amusing if you replaced “60%” with “0%”
)
October 31, 2011 at 7:40 am
this was the answer I came up with, and I believe is correct.
October 31, 2011 at 7:50 am
This is the correct answer.
0% as one of the answers would definetely make the question more interesting. That creates a paradox: The answer is than both right and wrong.
The interesting thing about this puzzle is that the answer influences the question.
October 31, 2011 at 7:55 am
yes! This is the correct answer!
October 31, 2011 at 8:25 am
More generally, this just comes down to the fact that it’s always possible to define a probability metric P on a discrete subset X of [0,1] where X is not {1} with the property P({x}) does not equal x for any x in X. Stated in this fashion, it seems much less paradoxical.
The problem as stated is solvable because when 0 is not an answer choice, it so happens that P({0})=0. When 0 is an answer choice but everything else stays the same, we have P({0})=.25 and hence lose the fixed point.
October 31, 2011 at 2:23 pm
Yes, exactly. It’s interesting that this is not obvious to more people (where by “obvious”, I mean, evident after lots of thought, of course!).
It’s a funny joke, though!
November 2, 2011 at 9:45 am
replacing “60%” with “0%” would not change anything, this would only eliminate the possible solution of 0%, leading to no possible correct answer…
For me this riddle is just a case of “catch 22…”
October 31, 2011 at 7:31 am |
There’s 3 answers but 4 choices. If a computer were to randomly choose 1, there’s a 50% chance it’ll choose 25% and a 50% chance it’ll choose either 50% or 60%.
I’m sure there’s a mathematical way to figure out the probability of a computer choosing the right one. I’ll figure it out and post it tomorrow if I remember.
October 31, 2011 at 7:32 am |
Reminds me of this, alhough not quite the same
October 31, 2011 at 9:45 am
This might be even funnier if they were Canadian, eh.
October 31, 2011 at 7:38 am |
I’m with repton on this. I think RW’s analysis is a bit off. If you randomly pick a) or d) you have to ask yourself if it’s the right answer. So is there a 25% chance of picking 25%? No, there’s a 50% chance. So a) and d) are not correct. Apply the same to b) – is there a 50% chance of choosing that answer? No. Same applies to c). As none of the options will give you the chance of having chosen that answer, then none is correct, so there is a 0% chance.
October 31, 2011 at 10:34 am
But 0% isn’t one of the options to pick so…
October 31, 2011 at 6:42 pm
Mick, there is no reason to assume that one of the options has to be correct.
November 1, 2011 at 6:42 pm
Josh, yes I know. My point is that 0% isn’t one of the options. There is no correct anwser to this.
October 31, 2011 at 7:42 am |
It’s a multiple choice question with 4 wrong answers. No one ever said that the question needs to have a right answer included in the 4 options. The answer stil exists, it’s just not one of the options.
Therefore if you select an answer randomly, you have a 0% chance of getting the right answer. 0% is also the answer to the question.
The puzzle is solvable, and the right answer is 0% chance. As far as I can tell it’s the only answer that won’t produce a loop or a paradox as well.
0% couldn’t be included as one of the four options though, or it would stop being the correct choice.
October 31, 2011 at 7:44 am |
The reason Richard is wrong is that the chance of getting any of these answers is not 33%. Imagine the following: What colour are zugs? a) yellow, b) blue, c) yellow d) red. If zugs are yellow, you have a 50% chance of being right. If zugs are blue or red you have a 25% chance of being right. The ambiguity in the original question is not because 33% is missing; it’s because the chance of getting 25% is 50% and vice versa.
October 31, 2011 at 1:47 pm
I’m with all of you saying that 33% is wrong. There’s a fifty percent chance of randomly choosing 25% and a 25% chance of randomly choosing either of the other answers. I’ll accept that 0% is correct even though it isn’t in the answers, because none of the given answers can ever be correct, as soon as you identify one as being correct, it becomes wrong.
October 31, 2011 at 6:24 pm
Zug being the German word for train, and trains in Germany predominantly being red, I’d have to say d) red.
October 31, 2011 at 7:46 am |
having two answers the same does not make the chances of picking the right one 33%. There is no correct answer because if you choose 25% there is a 50% chance of picking it, but if you choose 50% there is a 25% chance of picking it.
October 31, 2011 at 7:46 am |
There is a way to make both ‘a’ and ‘d’ be correct answers. If an eight sided die had ‘a’ and ‘d’ on one side each, then the answer “25%” has a 25% chance of being selected even though ‘a’ and ‘d’ each have a 12.5% chance of being selected.
October 31, 2011 at 7:49 am |
I’ll go with the interpretation that “choose an answer to this question at random” means that choosing among the four option tags, not as choosing among the distinct values.
Thinking about variants, in that case, if the 4 values are all distinct, then the probability of picking any is 25%, so if one of them is 25%, that is the answer.
If one of the values repeats (and none of the other distinct values are 25%), then if the repeating value is 50%, then that is the answer; rather either of the two tags are the answers.
similarly 75% occurring three times, or 100% occurring all four times have valid solutions. Other variants dont.
October 31, 2011 at 7:49 am |
I got 33.3%
There’s a 33.3% chance the answer will be 25%, and a 50% chance you’ll choose 25%
There’s a 33.3% chance you’ll choose 50% and a 25% chance you’ll choose 50% – and the same for 60%.
So 50% x 33.3% + 25% x 33.3% + 25% x 33.3% = 33.3%
Bit annoying there’s ‘no answer’. I thought I had it wrong all weekend. Chances are I still have it wrong, but I’d have liked to have been put out of my misery at least.
October 31, 2011 at 7:50 am
OOps – replied to the wrong post.
October 31, 2011 at 7:53 am
But 33% isn’t one of the four options, so you have no chance of selecting it as the answer randomly.
The puzzle is solvable, so Richard is wrong about this one. The answer is 0%
October 31, 2011 at 7:50 am |
I gave B as the answer. Ignoring the question, in any multiple-choice question you assume that one of the answers is going to be correct. As there are 4 possible answers, that means you have a 1/4 chance (25%) of getting it correct, regardless of the question. As there are two answers here that give the 25% option, that means you have twice as many chances of getting it correct i.e. 50%. So the answer is B, no?
a) 25%
b) 50%
c) 60%
d) 25%
October 31, 2011 at 7:59 am
No: because when you say that there is a 50% chance that you answer the question right, then the answers a, c and d are wrong.
The point is that the answers make each other incorrect. Just see it this way:
When you pick a, it changes the answer into “there is a 25% chance that you have this answer right”. But that is incorrect because there is a 50% chance that you pick 25%.
When you pick b, it says “there is a 50% chance that you have this answer right”. But that is incorrect because there is a 25% chance that you pick 50%.
When you pick c, it says “there is a 60% chance that you have this answer right”. But that is incorrect because there is a 25% chance that you pick 60%.
d is the same as a.
So all answers are incorrect making it 0%.
October 31, 2011 at 8:02 am
If the answers were this:
a) 25%
b) 50%
c) 0%
d) 25%
And you randomly choose c, it would say: “there is a 0% chance that you have this answer right”. But that is incorrect because there is a 25% chance that you pick 0%.
Again, making 0% the right answer. But according to the question 0% cannot be right, but it is. If you were not confused, then you are now.
October 31, 2011 at 7:50 am |
The answer is that you don’t have an answer? Uninspiring to say the least.
I agree with Repton above, it’s 0%.
Artificially restrictin the possible answers to a question to some small subset of logically possible answers doesn’t mean the correct answer will always be part of that subset. In this case it’s not, which can be easily shown using proof by contradiction:
Assume a) is correct: d) must necessarily also be correct, but then you’d have a 50% chance so they’re both false.
Assume b) is correct: answer is 50%, but only 25% of the answers are this.
Same logic for c) as for b)
The above is assuming that ‘random’ means a flat distribution, rather than some weighted function giving different chances of picking each answer. Anything else would be entirelu unfair if not specified in the puzzle, since you could play with the function to arbitrarily force any answer to be correct.
This isn’t really a puzzle, it’s just a question and four wrong answers.
October 31, 2011 at 7:52 am |
It depends which way you read the question. If you read the question as the answer being 25% chance of picking a random letter, then your chances of picking the 25% as the answer would be 50%. But if you did not take into consideration that 25% is the answer but any of them could have been the answer to something different, the right answer then it would be 30%. Does that make sense? I am confusing myself now. I thought it was 50% anyway.
October 31, 2011 at 7:53 am |
This very puzzle was brougth up in my mathematics class on friday.
First I reasoned as follows:
* If you choose 25 %, your answer is wrong, since the probability of chosing 25 % at random is 50%:
* If you choose 50 %, your answer is wrong, since the probability of chosing 50 % at random is 25%:
* If you choose 60 %, your answer is wrong, since the probability of chosing 60 % at random is 25%:
* Therefore there is no right answer and the probablity is 0%.
Then someone told me there was a sublte error in my premises, so I reasoned as follows:
* The prior probability of chosing any of the options at random is 25%
* The probability of chosing 25 % at random is 50 % since 2/4=0,5
* Therefore the right answer is option B 50 %
This was the consensus until somebody pointed out that this means 50 % IS your answer to the question, which means that WHAT IS BEING ASKED FOR is the probability of THIS answer.
So now I’m inclined to think my first response was correct.
October 31, 2011 at 8:05 am
Your first response is correct.
Because if 50% is right, because you have 2 answers of 25%, which can both be right, then 50% is also right. That adds up to 75%. That answer is not there. So: 0%.
October 31, 2011 at 8:18 am
Your first response is correct – the probability of choosing ’25%’ is 50%, because that is two of the four answers. But the question asks for the probability of randomly selecting the correct answer (i.e. the answer ‘x%’, for some ‘x’ which you have an x% chance of randomly selecting), not the probability of selecting 25% – as you have a 50% chance of selecting 25%, 25% cannot be correct.
October 31, 2011 at 7:59 am |
Well, I thought there wasn’t any answer to the question, since there is no question regarding the options.
“If you choose an answer to this question at random, what is the chance that you will be correct?” What is the question “this” is referring to? There is no previous question, so there is no answer.
October 31, 2011 at 12:06 pm
Why do you need a previous question? “This” usually refers to the current thing/place/time/whatever.
For example:
This sentence has seven words in it.
The “this” in that sentence refers to itself, just as the “this” in the question refers to the question.
October 31, 2011 at 8:04 am |
Well, for me the answer was 60%; maybe different for others if they chose different answer at random. I can only think it’s pretending to throw us, when actually it’s just asking a very simple question, and asking us to say out loud what we picked? If we play the puzzle logically, then we subvert the puzzle as we’re no longer picking an answer at random??
Any good?
October 31, 2011 at 8:08 am |
Bjarte, so are you saying that if you are being asked what the probability of you picking 50% is then wouldn’t that then be 25%?
October 31, 2011 at 8:10 am |
I say the correct answer is B, 50%. You have a 50% chance of choosing 25%, which is the correct answer.
Note, that if do you answer B, your answer really is wrong. But, nobody said that the answer you choose should be the correct one!
If one does require that the answer you give is also the correct one, then there is no solution to the problem. But, as said, this is not what is required in the formulation of the question, as far as I can see. I also think Bjarte above me was correct in his latter reasoning, and is being unnecessarily swayed by the popular vote.
)!
Though I bet many will disagree with me on this one, I’m sure (and there is always the possibility that I’m wrong — failure is always an option for me
October 31, 2011 at 8:19 am
Wait, no, darn. That doesn’t feel right. I gotta think this again.
October 31, 2011 at 8:28 am
Well of course the answer has to be correct. Otherwise it’s entirely meaningless to ask. You may as well state the the answer is ‘cheesecake’, and question why anybody was expecting you to look for the correct answer.
You are only answering “what is the probability that you choose ’25%’?”, which is not what the question asks for: assuming 25% to be correct leads to a contradiction (you do not have a 25% chance of choosing it), and thus 25% cannot be correct.
October 31, 2011 at 8:13 am |
It is interesting to see the number of people who think the text of the alternatives have a bearing on the probabilities of the question.
The classic riddle in this regard is “This sentence is false”. The solution, of course, is that the sentence is meaningless and has no truth value. It is just a sequence of letters.
Similarly, the options here were never stated to contain the possible probabilities to the question. The question was “what are the chances” and because the options contain something that looks like probabilities, people immediately assume that they must be. “This probability is false”.
October 31, 2011 at 8:17 am
Very good explanation, Anders. I also believe there really is no question to answer.
October 31, 2011 at 8:30 am
Actually, thinking a bit more about this, I’m guilty of an unstated assumption as well. The question never actually said that the choice was between the four letters, that was an assumption on my part based on years of conditioning.
The range of possible answers then is infinite, and the odds of hitting the right one when choosing at random is not quite 0 but infinitely close to it
October 31, 2011 at 8:18 am |
AnnieB: Yes, but then the probability of randomly picking that answer (the very thing that is being asked for) is not right according to the anwer you actually picked. :-S
October 31, 2011 at 8:27 am |
A fun variant:
If you choose an answer to this question at random, what is the chance that you will be correct?
a) 25%
b) 50%
c) 50%
d) 0%
October 31, 2011 at 8:27 am |
After initial thought that it’s 50% I realized that there’s no right answers therefore the chance to choose the right answer is 0%
That’s why I commented it as sneaky on Friday
October 31, 2011 at 8:38 am |
Assuming “correct” answer is also random, with probabilities of P(25%) = 1/2, P(50%) = 1/4, P(60%) = 1/4, then the probability for choosing the correct answer at random is 50%.
If you now allow your answer to retroactively change the “correct” answer, then the question becomes unanswerable, however, why would you do this? That makes no sense.
October 31, 2011 at 8:40 am |
This is actually an example of a Russel Paradox
The probability of choosing “an” answer is actually 25% since there are 4 possible options. However the likelyhood of picking an answer that lists as 25% is 50% (because the answer 25% is listed twice).
Now you would think that the answer would be 50%, but the chances of you picking that answer is only 25%. Which brings us back to the front
I can’t remember where but I’ve seen this puzzle before where this answer was given to the puzzle.
October 31, 2011 at 8:41 am |
It a trick question, so the chance of answering correctly is 0%
October 31, 2011 at 8:49 am |
Chris wrote: ‘Artificially restrictin the possible answers to a question to some small subset of logically possible answers doesn’t mean the correct answer will always be part of that subset. In this case it’s not, which can be easily shown using proof by contradiction:’
I’m with Chris on this. It’s a bit of an Alice-in-Wonderland-logic problem, but it seems to me that the answer must be 25%, irrespective of what each ‘answer’ reads as.
October 31, 2011 at 8:50 am
Or it’s 0%.
There – that helps the debate, dunnit
October 31, 2011 at 8:52 am |
Okay, so let’s rephrase the question. If you chose an answer to this question at random, what are the chances of you being correct:
a) Apple
b) Pear
c) Banana
d) Apple
October 31, 2011 at 8:59 am
Oh Christ, now I’m starting to see the logic of 33.33333% (as mentioned above).
October 31, 2011 at 11:02 am
Thats a good idea. I was also experementing in this way. But there is something to clarify at all: Does “chose at random” mean that we close our eyes and tip with a stylo on one of the four answers or does it mean that we read the answers an then make a well-considered choice ?
October 31, 2011 at 12:57 pm
How about this version:
Four envelopes contain the answers: 25%, 50%, 50%, 100%
I ask AnnieB to pick one envelope at random. What is the probability that AnnieB has picked the correct answer to this question?
October 31, 2011 at 9:08 am |
This is quite a stretch, but flip a coin to choose whether or not you answer the question. Then flip the coin twice to choose an answer. Then ‘a’ and ‘d’ could each be correct answers.
October 31, 2011 at 9:12 am
Not answering would have to count as not answering correctly.
October 31, 2011 at 9:32 am |
I feel Cheated by this. we have spent all weekend pondering what the answer would be only to find out there isnt one.
Rubbish!!!
October 31, 2011 at 6:16 pm
That’s not true, there is one, it’s 0%.
October 31, 2011 at 9:43 am |
I think AnnieB has decoupled the answers from the question. If the question is “What are the chances of you being correct?”, the answer cannot be “Apple”, or “Pear” or “Banana”. Therefore you would have a 0% chance of answering correctly.
In the taxonomy of word games, I would be inclined to class this as a Riddle rather than a Puzzle. Or maybe it would be better termed a Linguistic Illusion; a bit like Chomsky’s “Colourless green dreams sleep furiously”.
October 31, 2011 at 9:45 am |
Multiple choice questions are very common in the school system, so we’ve been conditioned to expect that one of the choices will always be a correct one. Think about a question like this one for example:
What is 2+2?
a) 1
b) 2
c) 3
d) 5
The question is still valid and answerable, even if none of the given options are right. Of course it’s not a fair question, but no one ever said it would be. You just expect it to be fair because that’s how it’s always been in the thousands of multiple choice questions you’ve completed in your life.
The question in the friday puzzle too is solvable and has a correct answer, which is 0%. It is not the equivalent of “this statement is a lie.” which is a true paradox.
October 31, 2011 at 10:21 am
It is the equivalent of the liar’s paradox in that people ascribe meaning and truth value where none is warranted
October 31, 2011 at 10:41 am
But the liar’s paradox doesn’t have a valid answer, while this question does. Just because you are prevented from giving the right answer, doesn’t mean that a question becomes unanswerable.
Of course you can get into semantics about what is the true meaning of words like chance, random, question, choose, answer or is. Human communication relies largely on certain unspoken assumptions. If you peel back enough layers, you can make any communication impossible.
October 31, 2011 at 11:25 am
The liar’s paradox does have a valid answer, and I gave it above. The correct answer to the liar’s paradox is that a sentence is just a string of letters, and to assume there is meaning or truth value is unwarranted.
(of course the answer to the real, original liar’s paradox, is that all Cretans are liars, is that not all Cretans are liars and the speaker is a liar – it’s not even a paradox, really)
You are correct that in normal conversation, we have to be able to rely on assumptions, but when those assumptions turn out to be wrong, we make mistakes, misunderstand, get things wrong, perhaps get into fights. That is the real point of riddles like the liar’s paradox. Challenging assumptions, understanding when we can, understanding when we have to.
In riddles though, assumptions are nearly always wrong. The riddle format is designed to exploit such things (the surgeon is a woman)
October 31, 2011 at 11:59 am
By that same reasoning we can declare half the riddles and puzzles in the universe to be unanswerable, because after all “a sentence is just a string of letters, and to assume there is meaning or truth value is unwarranted.”
Suppose a question asks you what is the shortest amount of time in which John and Erica can complete the chores, which was a recent weekly puzzle. Do you then give the answer that time is relative, and we need to know how fast the observer is moving or whether they’re near any highly massive objects? If you stretch enough, you can make almost anything become an answer to any puzzle, but what is the fun in that?
While assumptions are indeed dangerous, all of communication and intelligent thinking becomes worthless if no assumptions at all are made. You can always derail anything with the equivalent of “But what is Truth reallly?” A lot of the beauty in riddles comes from knowing which assumptions are appropriate for the situation. A puzzle can have multiple answers depending on which assumptions are made to begin with.
October 31, 2011 at 12:25 pm
Not really, you are being overly dramatic here. The assumptions in a riddle are stated in the question. This is always true, and in 99% of riddles, this is the entire point of the whole thing, and the riddle leaves something unsaid which you assume, and this throws you off.
By challenging assumptions and going strictly by what is in the riddle (or indeed exam question), we do not “declare half the riddles and puzzles in the universe to be unanswerable”, we in fact answer them correctly. The people who stay with their assumptions never understand the point of riddles and questions of this nature.
So, assume only that which is given in the question, nothing else. That works, assuming the riddler (or exam setter) is competent.
In this case, the question only asked for the odds of answering the question correctly. Nothing else. In every exam I have ever been in, and in every book of puzzles, there is *always* the statement that the answer is to be chosen from the alternatives listed. If there is no such statement, the question is open-ended and any answer may be given. Here, there were no such statement
The puzzle you refer to did leave many aspects open, which led to a healthy discussion about possible alternative answers allowed by the phrasing of the question. This is not wrong, it is the whole point.
October 31, 2011 at 12:30 pm
By the way, maybe you’ve missed this, but the people who claim there is no answer are the ones who go with the conventional assumptions (the percentages are probabilities, the answer is one of the 4 etc.). They say it can’t be answered. I actually gave an answer, with justification, and I would defend it to any mathematics professor. So please don’t claim that *I* would leave half the puzzles without an answer.
October 31, 2011 at 1:10 pm
My original comment was not aimed at you specifically, but rather at people calling the puzzle unanswerable, a waste of time, or even not a puzzle at all. I’d be interested in knowing what you did think of the puzzle overall?
I agree that an answer should be chosen from the alternatives given in a question like this. Are you saying that it needs to specifically stated in every single puzzle, or we can assume that any answer can be chosen?
The point was the the lack of a correct answering option does not make the question itself invalid. The existence of a solution is separate from your ability to give it as an answer. So 0% is still a valid answer to the question presented, even if its not included in the options. In fact it would seize being a valid option if it was included. That’s part of the beauty of the puzzle. To me the whole point of the puzzle was challenging those assumptions people make about the nature of multiple choice questions.
October 31, 2011 at 1:21 pm
I think it needs to be stated. At the very least something like “which of the following…” or similar. The version I just saw here in a comment, which rewrote it to have four envelopes and ask to pick one envelope at random would also be acceptable, and comes close to being the question most people here have answered.
I do believe however that the question becomes more interesting when you read it the way I did. Then it becomes about challenging assumptions, and not answering something that was never actually asked, and all the rest that I have argued above and elsewhere – as opposed to a not-really-answerable probability calculus exercise. It would be something I’d expect from a psychology professor, and more in line with the events, books and videos that have made Richard Wiseman famous. All these semi-skilled maths problems on this blog are so far removed from the image I have of him
October 31, 2011 at 10:21 am |
The question tells us that there is a correct answer by asking us what the chances of picking it are. The only way the answer can be 0% is if there isn’t a right answer. So 0% can’t be the right answer.
October 31, 2011 at 10:37 am
The question asks “What is the chance you will be correct?” It does not ask what is the chance of picking the correct answer. So it doesn’t tell us anything about the existence or non-existence of a correct answer.
October 31, 2011 at 10:47 am
Just because it tells you there’s a correct answer doesn’t mean there is one; the question might be wrong. Among the answers given, there’s certainly no correct one, and you absolutely have a 0% chance of the correct answer if you’re restricted to the selection of a, b, c, or d.
If you decide you’re not restricted to the answer’s suggested, then it becomes impossible to really quantify. My best answer if you’re *not* restricted to just a, b, c, or d is:
You then have an infinite choice of possible answers. If the selection is truly random, each is equally likely, and thus each has a probability of 1/(infinity), which is undefined but is smaller than 1/x for any positive x. And so whichever answer is correct, your chance of picking it basically end up at zero again (or more strictly, infinitesimal), and in this case I can’t see any knock-on effects where the definition of that answer as correct would lead to a contradiction.
Of course, to suggest we could truly pick at random from an infinite set is absurd.
October 31, 2011 at 10:47 am |
0% is the answer. If the answer is 25%, there is a 50% chance of picking it => contradiction.
If the answer is 50%, there is a 25% chance of picking it=> contradiction, similarly with 60%. There is a 0% chance of picking 0%, and this is the only consistent possibility.
October 31, 2011 at 10:53 am |
I agree that the correct answer is 0%, but it might be helpful to articulate the difference between the 25% chance of picking a given answer as cited by many correspondents and the 33% chance cited by RW.
Put simply, it depends on the algorithm selected by the chooser. Faced with the given list of options and told to pick one at random, there are two equally reasonable procedures which could be adopted. One is to ignore the content of the four options and simply roll a four-sided die and thus pick an answer at random. Only once a, b, c or d had been chosen would you pay any attention to the answer selected.
The other option, which RW must have had in mind, would be to assess the number of different answers first, ignoring how the options were labelled. There are three different answers, so to choose randomly amongst these, one throws a three sided die (or more practically a six-sided die) and thus picks an answer at random.
As far as I can see, both procedures are reasonable and with no other information there is no particular reason to prefer one to the other. This means that if the options were
a. 33%
b. 50%
c. 33%
d. 25%
Then 33% would arguably be a correct answer (to the nearest whole number).
October 31, 2011 at 11:09 am |
Wait a minute….. Richard gives us a problem he doesn’t think there is an answer to, leaving us to squabble amongst ourselves as to what the answer may be. Is this some kind of warped psychological experiment?
October 31, 2011 at 11:44 am
That would be funny! I like the comments of the one’s who feel all indignant, as though they’re actually owed an explanation. I would like to see it resolved as I have the small problem that I shared this puzzle with my FB friends and they are all waiting for the answer. Jokes on me! ROTFL
October 31, 2011 at 12:38 pm
Next Friday’s Puzzle:
“Will the number of indignant replies on Monday be greatest if I posted the correct answer to this question, if I posted the wrong answer to this question, or I posted no answer?”
October 31, 2011 at 11:20 am |
What a complete waste of time!!!
October 31, 2011 at 11:21 am |
What a complete waste of time!!!
October 31, 2011 at 12:14 pm |
It’s a paradox in that a question that doesn’t appear to have an answer, does have an answer (that is, 0%), but only because we’re applying the conventional rules of MCQ. Given that two choices are identical indicates that this isn’t a typical MCQ. This is a hint that since MCQ rules no longer apply, the question can be reinterpreted as: “If four envelopes contain answers: 25%, 25%, 50%, 60% and I ask someone to pick one at random, what is the probability that she has chosen the correct answer to this question?”
Since the question is no longer in MCQ form, the paradox is resolved — until I replace the 60% envelope with one conatining 0%, that is.
October 31, 2011 at 2:30 pm |
“If you roll a pair of dice at random, there are essentially 11 different outcomes, so the probability of each is 1/11.” We know this is wrong, because the randomisation procedure has been specified. The problem of the present question is that the randomisation procedure has not been specified. Are we to choose one of a, b, c, d at random, like wagdog? That is certainly the most common interpretation of the kind of words used here.
October 31, 2011 at 2:59 pm |
The question does not ask you to choose an answer from the choices below it. It only asks “what is the chance that you will be correct?” Therefore, the ABCD answers below the question are unrelated to the question itself, and the answer is 25%!
October 31, 2011 at 4:46 pm |
I’m not convinced humans are capable of making a random selection.
October 31, 2011 at 5:27 pm |
i suggest it’s a basic 1-in-4 chance and someone has changed the numbers from the original puzzle and failed to realise the impact it would have outcome. If you assume the normal multiple choice format (1 question & 4 possible answers and only one is the correct) then it’s always a 1-in- 4 chance of being randomly being correct answer
November 1, 2011 at 4:35 am
But that’s assuming it’s still out of four possible choices – which one can’t assume if you’re going to discard the provided choices.
October 31, 2011 at 6:28 pm |
I spent a good few minutes vacillating between 25% & 50% then smoke started coming out of my ears & I decided to wait for the answer. Strangely relieved that there isn’t one.
October 31, 2011 at 6:46 pm |
It took me some time but I think the right answer is 42.
October 31, 2011 at 6:52 pm |
I think this is a question about linguistics, the chance of you doing something is always the same, you either will or you won’t. When you have four answers, it is “probability” that will determine which you will pick or which is right.
October 31, 2011 at 9:43 pm |
I think its kind of like the Russel paradox. If the answer is 25% then beacause it appears 2 times you have 50% chance of choosing it. If the answer is 50% then there are only 1 value which appears 2 times therefore the answer is 25%. So the answer is 50% if and only if the aswer is 25% and vica versa.
October 31, 2011 at 10:40 pm |
Has anyone consulted Kurt Godel or Douglas Hofstader?
October 31, 2011 at 11:50 pm |
the answer is
e) 20%
October 31, 2011 at 11:57 pm
as clever as that pi puzzle
but more acceptable
November 1, 2011 at 4:12 am
No, the answer is f) 16 2/3 %
Wait, no, it’s g) 14 2/7 %
…No, wait, it’s….
November 1, 2011 at 5:18 am |
Of course the puzzle did not say that the answer had to be in the list, so 33% is perfectly acceptable.
November 1, 2011 at 5:48 am |
I am thinking like this:
If the answer is 25%, the chance is 50% – contradiction
If the answer is 50%, the chance is 25% – contradiction
If the answer is 60%, the chance is 25% – contradiction
Since none of those answers is correct, the chance is 0%, which is correct, because 0% is not one of the listed options.
November 2, 2011 at 4:17 pm
This is my reasoning, too. Tricky puzzle.
November 1, 2011 at 9:04 am |
Outright cheating! Best expressed by person who said “WTF?”
November 1, 2011 at 9:36 am |
The answer is ‘no’
November 1, 2011 at 12:03 pm |
a) I was right in thinking that I was wrong.
b) I was wrong in thinking I was right.
November 1, 2011 at 3:20 pm |
Any multiple choice question must always allow for an (n+1)th response of “none of the above”. That’s simply because in even the best prepped quiz misteaks can creep in.
Thereforewise, is five possible answers: a, b, c, d, e, none,
If I guess one of those with equal randomness, then in one occassion in five (1 in 5) I will guess none.
As none of the answers in Richard’s quiz is right, then none is right.
Means I am guessed right 1 in 5. Which is (when multiplied using simple school arithmetic) 4 in 20 or 20%
Thereas, 20% is the rightestmost answer in this case.
And here I rest that case.
November 1, 2011 at 4:17 pm |
Considering the many-worlds interpretation, it’s you have a 100% chance to choosing the right answer somwhere.
November 3, 2011 at 4:08 pm |
The question does not speciifcally limit the correct answer to the choices listed below. In fact, it suggets quite the opposite. A legitimate answer might – in fact – be “elephant.”
November 6, 2011 at 3:47 pm |
Suppose that “random” does not refer to a uniform distribution. Then I can pick (a) and (d) with 1/8 probability each, and spread out the remaining 3/4 on alternatives (b) and (c) equally. Then I will be correct with 25% probability.
You can do the same to justify answer (b) and (c) too.
Another solution suggested to me as that one may choose two distinct alternatives (as is common in many multiple choice problems). Then there is a 25% chance of choosing (a,d).
January 31, 2012 at 4:15 am |
At first I broke it down into 33% for each actual value.. but even if 33% was an option it would change the actual true answer to 100%.. I think the key to the puzzle is the word RANDOM.. if all the question asks is for one to RANDOMLY choose an answer then none of the choices can be wrong
February 21, 2012 at 1:37 am |
The answer is 33%.
Reasons: The question isn’t defined, just need to work out the ‘chance’ that one of those answers is the correct answer. The chance you will pick the correct answer at random is 33%
Forget the actual choices, they are irrelevant and only there to confuse. Another example:
Pick an answer at random, what is the chance that it is correct:
A) Sausage
B) Tree
C) House
D) Sausage
The chance is 33%, the question isn’t even important, because the answer is random.
February 21, 2012 at 1:52 am
Just to clarify again, the question isn’t important, the value’s of the 4 answers isn’t important either.
You chose either A, B, C or D. Because 2 of them are the same, the probability you pick the right answer is 33%.
The answer to this puzzle doesn’t have to be one of the value’s of ABCD. My above example doesn’t have a “Sausage Percent” of being correct lol.