| Nice! What's really fun about this problem is that you can have very convincing arguments for 1/2 being the correct answer, and very convincing arguments for 1/3 being the correct answer. And for either you can make subtle reformulations that supposedly illustrate how ridiculous this answer is. And there is no way to know. There is no gold standard for designing an experiment that would show whether 1/2 or 1/3 is correct. You could set up something that generates millions of pairs of (virtual) kids and then count the pairs that fit. But each of these experiments will have built-in the assumption on which the response is ultimately already predicated on. The only thing really convincing would be if everybody, all "sides", could agree on an experiment with an outcome that they would feel bound to. Then one could settle this once and for all, whether it's 1/2 or 1/3 or 13/27 or 729/1459 or whatnot. But people will never agree on such an experimental setup. Which tells me that this is not a mathematical problem. This problem is either underspecified or it's contradictory. If it was uniquely specified then we could just use probability theory with its axioms and inference rules to derive at the correct answer. But we obviously can't, since nobody can agree on how to formally note this down. |
You’re right.
I wrote in another comment the solution down to this two unspecified elements:
P(you tell me that you have two children including at least one boy | you have two boys)
P(you tell me that you have two children including at least one boy | you have one boy and one girl)
If one assumes that they are equal (why?) the answer is 1/3.
If one assumes that the latter is half as probable the answer is 1/2.
Whatever the assumption that one finds more natural the point is that an assumption is needed.