I just read this interesting xkcd strip:

On a scale of 1 to 10, how likely is it that this question is using binary?

At first I thought it was funny, but as I got to ruminate a little over it, I was surprised to be unable to find an answer. As Karolis Juodelė pointed out, the probability is ε, as there is an infinite number of bases containing 1 and 10.

However, to get a finite answer, we can modify the puzzle like this:

On a scale of 1 to 10, how likely is it that this question is using binary vs. decimal?

So my question is: How should I solve this puzzle? Is there a correct answer at all? Is this what we call a self-reference paradox, like Multiple-choice question about the probability of a random answer to itself being correct?


closed as off-topic by ShreevatsaR, Jack D'Aurizio, Michael Albanese, Erick Wong, Grigory M Jan 12 '14 at 8:52

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  • "This question is not about mathematics, within the scope defined in the help center." – ShreevatsaR, Jack D'Aurizio, Michael Albanese, Erick Wong
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    $\begingroup$ This is not a puzzle. There is no method of evaluating the likelihood of things happening in other peoples heads. If you just want to get the right base, say $1$. $\endgroup$ – Karolis Juodelė Jan 11 '14 at 9:54
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    $\begingroup$ Actually, if you assume that all possible interpretations of $1$ and $10$ are equally likely, the probability is $0$, because all bases have $1$ and $10$. This could be base $2$, base $10$ or base $123456$. $\endgroup$ – Karolis Juodelė Jan 11 '14 at 9:57
  • $\begingroup$ Base $10$, definitely base $10$, all number bases are base $10$ $\endgroup$ – peterwhy Jan 11 '14 at 10:28
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    $\begingroup$ She doesn't know what a 4 is, so it is either binary, ternary or base-4. (It can't be unary because 10 is a number). $\endgroup$ – Empy2 Jan 11 '14 at 15:24
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    $\begingroup$ @anustart in your modified version, if your question is in binary (the word in common sense), then I don't understand your "2" there, but still your question is in base 10. Alternatively, if your question is in, say, tridecimal (the word in common sense), then your question is not in base 2 but in base 10. So I will definitely vote for base 10, like I said above. $\endgroup$ – peterwhy Jan 11 '14 at 18:37

(If I understand correctly, $10$ means certainly in binary and $1$ means certainly in decimal)

To answer your modified question, if I were to think like in your last comment (2014-01-11 20:48:42Z), then I would answer:

As likely as $1+\dfrac{10-1}{1+1} = \dfrac{11}{1+1}$

But for me, I choose to answer

As likely as $1+1+\dfrac{1+1-10}{(1+1)^{1+1+1}}$

  • $\begingroup$ If the question is decimal, the scale is 1 to 10, which means 1 = never binary and 10 = always binary, and also 5.5 will be a 50/50 chance. If the question is binary, the scale is 1 to 2, which means 2 = always binary and 1.5 (or 1.1 as binary) will be a 50/50 chance. Therefore I'd choose to answer "If decimal 5.5, if binary 1.1". Is there a single answer that will be correct independent on its base? $\endgroup$ – osvein Jan 12 '14 at 7:15
  • $\begingroup$ How about (b+1)/2 where b is the base? $\endgroup$ – osvein Jan 12 '14 at 7:23
  • $\begingroup$ That is equivalent to the first answer I gave above: $\frac{11}{1+1}$. $\endgroup$ – peterwhy Jan 12 '14 at 7:25
  • $\begingroup$ Ok, then I just realized the if the question is binary, the answer is binary as well. +1! $\endgroup$ – osvein Jan 12 '14 at 7:28
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    $\begingroup$ But then, why assign 50:50 chance to 1 and 10? If the question is in decimal, then 1, if binary then 10. $\endgroup$ – peterwhy Jan 12 '14 at 7:37

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