Multiple Choice Probability Puzzle 


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A MultipleChoice Probability Puzzle  2011/11/15Recently the following puzzle was running around the 'net:
So what is the right answer? Lots of people tweeted me about this, asking what the right answer is, and commenting that their brains were melting, exploding, or otherwise acting in an unpleasant or uncomfortable manner.
Similar "uneven" choices can be made to work in the other cases, and it's interesting (and difficult) to explore the possibilities. I'll leave it as an exercise for the interested reader to find distributions that make any given answer "right." There's even a way to make three of the options right all at the same time, although that really doesn't sound right at all! It works, although some people feel that it's cheating, and somehow wrong, not to choose among the options with equal probabilities. So here's another answer. Since none of the options seem to work, then none of them can be right. Since you will choose one of them, you will never get the right answer. Hence the probability of being right is 0. That option doesn't appear, so if you pick randomly you won't get it, and so your chance of being right is 0%, and so it's consistent. So that's another possible answer.
Now the second suggested solution  the 0% solution  doesn't work, because if we choose uniformly at random there is a nonzero chance that we pick that. So we're back to the nonuniform choice idea, and the many possibilities afforded by it. But underneath all this is another question: Why should the question have a "right" answer at all? So many of the tweets I got were indignant that there didn't seem to be a single, clear, correct answer. But life's not like that. Some questions don't have answers, and it's especially difficult when things are selfreferential. We can create impossible situations. People call them paradoxes, but the term is abused and overused. In some cases it's better simply to say that it's illposed, undefined, or otherwise impossible. Questions like that are great for exploring the limits of logic and the recesses of reason. And they make my brain hurt. That's their job. Other examples of paradoxes (or questions that are illposed, or something) include:
Comments
@snapey1979 said: Nice blog. Just because a problem is easily stated,
does not mean it is logically consistant, well posed, or soluble. That pretty much sums it up, and is related to the "2=4" proof here:
I've decided no longer to include comments directly via the Disqus (or any other) system. Instead, I'd be more than delighted to get emails from people who wish to make comments or engage in discussion. Comments will then be integrated into the page as and when they are appropriate. If the number of emails/comments gets too large to handle then I might return to a semiautomated system. We'll see.

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