I agree that George definitely does know more information overall, since he can concentrate his probability mass more sharply over the 4 hypotheses being considered, but I'm fairly certain you're wrong when you say that Sarah's distribution is 0.33-0.33-0-0.33. I worked out the math (which I hope I did right or I'll be quite embarassed), and I get 0.25-0.25-0-0.5.
Good point. I was treating the description of Sarah's encounter with the man as a proxy for "Sarah knows one of the man's children is a boy, but not which one." That seems to be the way it's usually intended when the problem is presented, but you're right that in the problem as described, Sarah has an additional relevant piece of information -- that the man is out with a boy. I think this is an unintended artifact of the way the problem is presented, though. The people presenting the problem are usually trying to get at something different. The usual intent of the puzzle is captured by "Sarah knows that one of Brian's two children is a boy, and George knows that his eldest child is a boy. What are the probabilities according to Sarah and George that Brian's other child is a boy?".
I think your analysis in terms of required message lengths is arguably wrong, because the purpose of the question is to establish the genders of the children and not the order in which they were born. That is, the answer to the question "What gender is the child at home?" can always be communicated in a single bit, and we don't care whether they were born first or second for the purposes of the puzzle.
Again, I think this is an unintended artifact of the way the puzzle is stated. The fact that Sarah sees one of the kids and doesn't see the other one gives her a way of individuating the kids other than their birth order. If we don't assume she has this method of individuation (as in the restated puzzle above) then the birth order is relevant.
I think we're in agreement then, although I've managed to confuse myself by trying to actually do the Shannon entropy math.
In the event we don't care about birth orders we have two relevant hypotheses which need to be distinguished between (boy-girl at 66% and boy-boy at 33%), so the message length would only need to be 0.9 bits#Definition) if I'm applying the math correctly for the entropy of a discrete random variable. So in one somewhat odd sense Sarah would actually know more about the gender than George does.
Which, given that the original post said
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Hi, I am relatively new to this site, I am not sure if this is the right place to be posting.
I am sure many of you are familiar with the following probability riddle:
"Sarah is walking along the street when she encounters a man. With the man is his son. He tells Sarah that he has only one more child at home. She is asked, 'what is the probability that my child is a girl?'"
Since Sarah does not know whether the boy is the elder or younger sibling, she needs to take four possible states into account. The father either had:
1) a boy, then a girl
2) a girl, then a boy
3) two girls
4) two boys
Since 3 is impossible (Sarah knows there is at least one boy) that leaves three options. Two of those options imply a girl, the other implies a boy. Therefore, she can conclude that her probability estimate must be that it is 66.6% likely that there is a girl at home, and 33.3% likely that there is a boy.
Compare this to George's situation.
"George is walking along the street when he encounters a man. With the man is his son. He tells George that the boy with him is his oldest son, and that he has only one more child at home. He is asked, 'What is the probability that my child at home is a girl?'"
George's probability estimate is clear: either the man had a boy then a girl, or he had two boys. Therefore, it is 50% likely that the child at home is a girl.
My problem is this: I understand probability exists in the mind. The actual answer to the question is 100% one way or the other. Still, it seems like Sarah knows more about the situation, where George, by being given more information, knows less. His estimate is as good as knowing nothing other than the fact that the man has a child which could be equally likely to be a boy or a girl.
If the reply is something like "Well, Sarah actually knows less so her estimate is less likely to be right" then that is something she could have figured out on her own, and then realized that assigning probability .5 is best anyways. That seems wrong.
I know I must be making a mistake somewhere: why does it seem like George learns less by knowing more?
Thank you for your help.