1
$\begingroup$

Possible Duplicate:
Card doubling paradox

I don't know how to resolve the following paradoxon: Assume two finite sets A, B, one being twice as big than the other, but you don't know which. You want to guess the bigger one. You want to choose A, but before that, you reason like this: "With probability 0.5 the other set B has half as many elements as A, and with probability 0.5 it has twice as many, so B's expected size is 1.25 |A|, so I should choose B."

What's the mistake in this line of reasoning?

  • 0
    I think it's because if $A$ and $B$ are already given, the probability that $B$ has twice as many elements or half as many elements of $A$ is either $1$ or $0$ since it's already determined.2011-04-02
  • 0
    Can this really be all that is to say?2011-04-02
  • 0
    The problem (from my view) is that you're assuming there exists a well-defined probability distribution on the set of possible sizes of $A$ and $B$ satisfying certain properties, and this is false. This is known as the two envelopes problem; see the question I've closed this question as a duplicate of or the Wikipedia article for details. The related discussion at http://mathoverflow.net/questions/9037/how-is-it-that-you-can-guess-if-one-of-a-pair-of-random-numbers-is-larger-with-p might also be helpful.2011-04-02
  • 0
    @Qiaochu: Should I delete my question, what would you give as an advice?2011-04-02
  • 0
    we generally don't delete duplicates. I think it's good to have more search terms pointing to the same question.2011-04-02

0 Answers 0