# Probability Problem. **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Created:** [December 31, 2014, 9:17pm UTC](https://boards.straightdope.com/t/probability-problem/708645 "2014-12-31T21:17:04Z")\ **Posts on this page:** 18\ **Page:** 1
**Author:** ![davidm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/davidm/32/225_2.png) [@davidm](https://boards.straightdope.com/u/davidm)\ **Post date:** [December 31, 2014, 9:17pm UTC](https://boards.straightdope.com/t/probability-problem/708645/1 "2014-12-31T21:17:04Z")
I already know what I think about this. I’m curious to see what others say. If you were to randomly choose one of the four answers below, what is the probability that your pick would be the correct answer to this question? A. 25% B. 50% C. 0% D. 25% ---
**Author:** ![Zebra](https://avatars.discourse-cdn.com/v4/letter/z/c0e974/32.png) [@Zebra](https://boards.straightdope.com/u/Zebra)\ **Post date:** [December 31, 2014, 9:20pm UTC](https://boards.straightdope.com/t/probability-problem/708645/2 "2014-12-31T21:20:02Z")
B 25% is correct but there are two correct and two incorrect responses. So 50% chance you would respond correctly. ---
**Author:** ![Smeghead](https://avatars.discourse-cdn.com/v4/letter/s/f1d935/32.png) [@Smeghead](https://boards.straightdope.com/u/Smeghead)\ **Post date:** [December 31, 2014, 9:37pm UTC](https://boards.straightdope.com/t/probability-problem/708645/3 "2014-12-31T21:37:34Z")
Gödel would like to have a word with you. ---
**Author:** ![davidm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/davidm/32/225_2.png) [@davidm](https://boards.straightdope.com/u/davidm)\ **Post date:** [December 31, 2014, 9:39pm UTC](https://boards.straightdope.com/t/probability-problem/708645/4 "2014-12-31T21:39:37Z")
> [@Smeghead](#): > > Gödel would like to have a word with you. Correct ---
**Author:** ![davidm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/davidm/32/225_2.png) [@davidm](https://boards.straightdope.com/u/davidm)\ **Post date:** [December 31, 2014, 9:42pm UTC](https://boards.straightdope.com/t/probability-problem/708645/5 "2014-12-31T21:42:30Z")
> [@Zebra](#): > > B > > 25% is correct but there are two correct and two incorrect responses. So 50% chance you would respond correctly. No. If that were correct then, since only one of the choices says 50%, your chance of picking it is only 25% so you have a contradiction. ---
**Author:** ![buddy431](https://avatars.discourse-cdn.com/v4/letter/b/6a8cbe/32.png) [@buddy431](https://boards.straightdope.com/u/buddy431)\ **Post date:** [December 31, 2014, 9:52pm UTC](https://boards.straightdope.com/t/probability-problem/708645/6 "2014-12-31T21:52:08Z")
What? [Self reference can lead to paradoxes?](http://en.wikipedia.org/wiki/Category:Self-referential_paradoxes) Who would have guessed. [Let R be the set of all sets that are not members of themselves](http://en.wikipedia.org/wiki/Russell%27s_paradox). Does R contain itself? ---
**Author:** ![davidm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/davidm/32/225_2.png) [@davidm](https://boards.straightdope.com/u/davidm)\ **Post date:** [December 31, 2014, 9:55pm UTC](https://boards.straightdope.com/t/probability-problem/708645/7 "2014-12-31T21:55:24Z")
> [@buddy431](#): > > What? [Self reference can lead to paradoxes?](http://en.wikipedia.org/wiki/Category:Self-referential_paradoxes) Who would have guessed. > > [Let R be the set of all sets that are not members of themselves](http://en.wikipedia.org/wiki/Russell%27s_paradox). Does R contain itself? Obviously you’re right. I wanted to see what kinds of answers and justifications for those answers that people would come up with. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [January 1, 2015, 1:00am UTC](https://boards.straightdope.com/t/probability-problem/708645/8 "2015-01-01T01:00:44Z")
We did this about 6 months ago. I’m not having success searching because I’m lazy and/or can’t find unique enough terms to get a small enough sample set to scan manually. But it is here to be found. ---
**Author:** ![Zebra](https://avatars.discourse-cdn.com/v4/letter/z/c0e974/32.png) [@Zebra](https://boards.straightdope.com/u/Zebra)\ **Post date:** [January 1, 2015, 7:05pm UTC](https://boards.straightdope.com/t/probability-problem/708645/9 "2015-01-01T19:05:55Z")
> [@davidm](#): > > No. > > If that were correct then, since only one of the choices says 50%, your chance of picking it is only 25% so you have a contradiction. Ultimately I knew that. But the chance of you picking 25% was 50%. 25% is correct and you have a 50% chance of picking that. You basically asked two questions. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [January 1, 2015, 10:11pm UTC](https://boards.straightdope.com/t/probability-problem/708645/10 "2015-01-01T22:11:33Z")
The question is not well-posed. What do you mean by “correct answer”? Is it the correct number (in this case 50%) or the correct “choice” in which case it is A or D. ---
**Author:** ![davidm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/davidm/32/225_2.png) [@davidm](https://boards.straightdope.com/u/davidm)\ **Post date:** [January 2, 2015, 1:50am UTC](https://boards.straightdope.com/t/probability-problem/708645/11 "2015-01-02T01:50:11Z")
> [@Zebra](#): > > Ultimately I knew that. But the chance of you picking 25% was 50%. 25% is correct and you have a 50% chance of picking that. You basically asked two questions. There’s only one question. _If_ you were to randomly choose, what is the probability that you would choose the correct answer? I’m not asking you to make a random choice, I’m asking you what the odds are that a random choice (if it were made) would be the same answer as if you consciously picked the correct answer. Of course, what happens is any choice you make leads to a contradiction. So the question is unanswerable. ---
**Author:** ![Princhester](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Princhester](https://boards.straightdope.com/u/Princhester)\ **Post date:** [January 2, 2015, 1:52am UTC](https://boards.straightdope.com/t/probability-problem/708645/12 "2015-01-02T01:52:05Z")
> [@davidm](#): > > If you were to randomly choose one of the four answers below, what is the probability that your pick would be the correct answer to this question? What question? The trick here is that you haven’t actually set any question. The term “this question” uses a pronoun - “this” - which would need to be defined to be meaningful, but nothing in the balance of your sentence is a question: - the words “If you were to randomly choose one of the four answers below” are not a question; and - the words “what is the probability that your pick would be the correct answer to” are not a question. The trick works because people read the sentence and jump to conclusions about the question being asked, and then confuse themselves because different possible questions give different answers. If the question was actually defined, the answer would be clear. ---
**Author:** ![davidm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/davidm/32/225_2.png) [@davidm](https://boards.straightdope.com/u/davidm)\ **Post date:** [January 2, 2015, 1:55am UTC](https://boards.straightdope.com/t/probability-problem/708645/13 "2015-01-02T01:55:49Z")
The question has one meaning. It’s not about tricking people. It’s about self-reference. There is no correct answer, but that’s because of the self-reference, not about any tricky wording. ---
**Author:** ![Princhester](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Princhester](https://boards.straightdope.com/u/Princhester)\ **Post date:** [January 2, 2015, 2:45am UTC](https://boards.straightdope.com/t/probability-problem/708645/14 "2015-01-02T02:45:28Z")
> [@davidm](#): > > The question has one meaning. It’s not about tricking people. It’s about self-reference. There is no correct answer, but that’s because of the self-reference, not about any tricky wording. What question? Write out the question for me without using a pronoun. There is no question. Every completed question in English can be written without using any pronouns. Let me have it. If you can’t, it wasn’t a completed question. ---
**Author:** ![Princhester](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Princhester](https://boards.straightdope.com/u/Princhester)\ **Post date:** [January 2, 2015, 2:51am UTC](https://boards.straightdope.com/t/probability-problem/708645/15 "2015-01-02T02:51:57Z")
This is self reference: _How many letters are there in this sentence?_ It’s a perfectly valid question and there is no difficulty answering it. This is not completed question: _If you were to randomly choose one of the four answers below, what is the probability that your pick would be? What is the correct answer to this question?_ My very slight and exact rephrasing shows that your original question is nonsense. There is no question. It isn’t completed. Be what? ---
**Author:** ![Princhester](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Princhester](https://boards.straightdope.com/u/Princhester)\ **Post date:** [January 2, 2015, 3:05am UTC](https://boards.straightdope.com/t/probability-problem/708645/16 "2015-01-02T03:05:04Z")
Alternatively, (sorry about the triple post) you could attempt to rephrase it as follows, inserting the original sentence in place of “this”, in an infinite but always incomplete regress: _If you were to randomly choose one of the four answers below, what is the probability that your pick would be the correct answer to [the question of "If you were to randomly choose one of the four answers below, what is the probability that your pick would be the correct answer to [the question of "["If you were to randomly choose one of the four answers below, what is the probability that your pick would be the correct answer to [the question of "If you were to randomly choose one of the four answers below, what is the probability that your pick would be the correct answer to [the question of…" **and so on ad infinitum** _ The problem here is that the series can never complete because there is no question so the self reference is to something that doesn’t exist. ---
**Author:** ![MaxTheVool](https://avatars.discourse-cdn.com/v4/letter/m/9fc29f/32.png) [@MaxTheVool](https://boards.straightdope.com/u/MaxTheVool)\ **Post date:** [January 2, 2015, 6:37am UTC](https://boards.straightdope.com/t/probability-problem/708645/17 "2015-01-02T06:37:51Z")
I don’t think it’s that far off from being a rigorously phrased problem… namely something like: Is there a positive integer n such that the the probability of randomly selecting n out of the list (0, 25, 25, 50) is n/100? If so, what is n? And the answer is, no n satisfies that condition. On the other hand, if the list was (50, 3432442), then 50 would be the correct answer, as it would also be the for the list (50, 50, 3, 3), and for the list (100, 100), both 0 and 100 are solutions. ---
**Author:** ![Princhester](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Princhester](https://boards.straightdope.com/u/Princhester)\ **Post date:** [January 2, 2015, 7:00am UTC](https://boards.straightdope.com/t/probability-problem/708645/18 "2015-01-02T07:00:01Z")
Indeed it’s the very fact that the sentence sounds close to being a meaningful question that makes people think it is one.