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.
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. You have to send >1 bit to Sarah only if she actually cares about the order of their births (And specifically, your “1 or 2 bits, depending” result is made by assuming that we don’t care about the birth order if they’re boys. If we care whether the boy currently out walking is the eldest child regardless of the other child’s gender we have to always send Sarah 2 bits).
Another way to look at that result is that when you simply want to ask “What is the probability of a boy or a girl at home?” you are adding up two disjoint ways-the-world-could-be for each case, and this adding operation obscures the difference between Sarah’s and George’s states of knowledge, leading to them both having the same distribution over that answer.
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
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.
Pragmatist is correct, I did not realize that the way I stated the problem was different than the original.
I full understand the solution to this problem.
However, lets look at the original problem. John only knows that one of the man’s children is a boy:
1) B, G | 0.33
2) G, B | 0.33
3) G, G | 0.00
4) B, B | 0.33
P(B)|(4) = 1 P(G)| (1,2) = 1
P(B)= .33 P(G) = .66
So lets say that now the woman tells John that the boy is also the eldest:
1) B, G | 0.5
2) G, B | 0.0
3) G, G | 0.0
4) B, B | 0.5
P(B)|(4) = 1 P(G)| (1) = 1 P(B)= .5 P(G) = .5
At first I saw a problem because John obviously knows more given the second piece of information, so the fact that his estimate is worse seemed really weird. What I think is going on here is that his learning more really does decrease his ability to predict the gender of the other child: Before, he had 3 options, 2 of which contained a girl-answer. Now, one of those 2 answers are taken away, so he currently has 2 options, 1 of which contains a girl-answer. As he becomes more informed about the total state of the world, his ability to predict this particular piece of information decreases.
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.
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. You have to send >1 bit to Sarah only if she actually cares about the order of their births (And specifically, your “1 or 2 bits, depending” result is made by assuming that we don’t care about the birth order if they’re boys. If we care whether the boy currently out walking is the eldest child regardless of the other child’s gender we have to always send Sarah 2 bits).
Another way to look at that result is that when you simply want to ask “What is the probability of a boy or a girl at home?” you are adding up two disjoint ways-the-world-could-be for each case, and this adding operation obscures the difference between Sarah’s and George’s states of knowledge, leading to them both having the same distribution over that answer.
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?”.
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
may not actually be implausible. Huh.
Pragmatist is correct, I did not realize that the way I stated the problem was different than the original.
I full understand the solution to this problem.
However, lets look at the original problem. John only knows that one of the man’s children is a boy:
1) B, G | 0.33
2) G, B | 0.33
3) G, G | 0.00
4) B, B | 0.33
P(B)|(4) = 1 P(G)| (1,2) = 1
P(B)= .33 P(G) = .66
So lets say that now the woman tells John that the boy is also the eldest:
1) B, G | 0.5
2) G, B | 0.0
3) G, G | 0.0
4) B, B | 0.5
P(B)|(4) = 1 P(G)| (1) = 1
P(B)= .5 P(G) = .5
At first I saw a problem because John obviously knows more given the second piece of information, so the fact that his estimate is worse seemed really weird. What I think is going on here is that his learning more really does decrease his ability to predict the gender of the other child: Before, he had 3 options, 2 of which contained a girl-answer. Now, one of those 2 answers are taken away, so he currently has 2 options, 1 of which contains a girl-answer. As he becomes more informed about the total state of the world, his ability to predict this particular piece of information decreases.
The fact that John predicts 0.5 while Sarah predicts 0.66 doesn’t mean that Sarah’s prediction is somehow better.