02
Nov
11

Spoiler warning

So this image has been floating around the big truck:

“If you choose an answer to this question at random, what is the chance you will be correct?

a) 25%
b) 50%
c) 60%
d) 25%”

It’s meant to be a gotcha, based on the reader’s (presumed) assumption that choosing an answer at random means each of the four answers is equally likely to be chosen.  That gives a 1 in 4 chance of getting the “right” answer — but both a) and d) correspond to that probability, which means that there’s a 2 in 4 chance of getting an answer of 25%… which corresponds to the answer in b), but there’s only a 1 in 4 chance that you’ll select b).  There’s no steady-state solution to this iterative process, so most people will mull it over and conclude “brain go splodey”, while a few math and comp-sci nerds will recognize a self-referential system and jump immediately to Russell and Gödel.

There is, of course, a satisfyingly smart-assed approach, which relies upon the fact that random numbers are chosen from some sort of distribution.  This remains unspecified in the question.  If we stipulate that random answers are distributed such that either b) is chosen 50% of the time or c) is chosen 60% of the time, we can make either one of those the correct answer.

Stupid math tricks: always fun on the internet.

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6 Responses to “Spoiler warning”


  1. 1 Woof
    November 2, 2011 at 13:18

    Isn’t it just all-Cretans-are-liars, obfuscated enough that it’s not immediately obvious? NTTAWWT, obv.

    • November 2, 2011 at 13:34

      Assuming a uniform distribution of random answers, yeah, it’s the same kind of paradox: you alternate between two incompatible truth assignments, each of which implies the other.

  2. 3 Erik
    November 2, 2011 at 14:30

    Another smart-ass response that I came up with:
    0%

  3. November 2, 2011 at 14:54

    Another smart-ass response would be that they are asking a human to choose a random value, not a computer. Humans are pretty bad at choosing random numbers; I’m not sure how bad they are in numeric terms, but a & d would most likely be chosen much less, and c much more. It’s not impossible that a & d would between them be chosen 25% of the time, or c would be chosen 60% of the time.

    • November 2, 2011 at 15:04

      Which gets back to the “didn’t specify a distribution” dodge (or insight).

      IIRC students tend to be biased slightly towards answering c) on four-option multiple-choice questions, but I very much doubt they’re 60% biased. A quick google didn’t turn up anything I could find on the subject before I lost interest, but I lost interest pretty quickly.

  4. 6 TMI
    November 2, 2011 at 19:46

    When this hit, I asked, “What would be a significantly different answer from any other?”
    .


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