Quote:
Originally Posted by Philo
Right, but why would there be any reason to make the assumption that that's the way the information was generated in the first place?
Is the problem somehow ill-defined unless we specify a way in which the information was generated?
I guess that's the idea. Although it seems somewhat telling that the examples they give produce the answer of 1/2 which so many people think it should be for the wrong reasons, i.e. nothing to do with the idea that it matters how the information was generated. With such a biased method of generating the information you can produce any probability from 1/3 to 100%
if the method is known, or if the experiment is repeated numerous times to reveal the bias. I suppose it's accurate to say the question is not perfectly specified but it seems rather pointlessly nitty to me. I suspect you could nit up most probability word problems in similar ways. (e.g. How do they know the coin is fair? or How do you know they don't only tell you the first coin came up heads when they know the second coin came up heads as well? How did they get this family to begin with?)
I suppose this question of how the information is generated has some motivation in that there are other versions of this puzzle where there is additional information in the puzzle about how you conclude "at least one child is a boy" and the way the puzzle reveals that information is relevant to the calculated probability. But in those cases the additional information is available to you.
After all, probability models are all about handling
incomplete information. The probabilities they generate will generally not be proved by frequencies for repetitions of an experiment about which the model has incomplete information. If the model had a perfect specification for the experiment the model would produce exactly one outcome with probability 100% (barring quantum effects).
If the
only available information is "at least one boy" then it seems to me the only reasonable way to update the probability model based on
the available information is to rule out GG leaving you with the equally likely outcomes of BB, BG, GB. That's certainly what a Bayesian would do looking at probability as an information theory and probability models as encoding the
available information.
PairTheBoard