The Florida problem did come from Mlodinow's book, but it was more like this: You recall that a distant relative has two children. You also recall that one is a one is a girl with an unusual name, like "Florida." But you don't recall of both are girls. What is the probability that both are girls?
There are several problems with Mlodinow's answer. Which was "A little less than 1/2" in the book, and (2-f)/(4-f) in reactions to the book where f is the frequency of the name "Florida." First, the original Two Child Problem was two problems: one with "The elder is a boy" and one with "At least one is a boy." The difference is that the first identified a specific child, so the answer was 1/2. Well, Mlodinow identified a specific child, didn't he?
And if you don't think he did identify one, note that the "elder child" establishes an order. The 1/2 answer applies with any kind of order. Like putting whichever child was more memorable to Leonard Mlodinow first.
Mlodinow also allowed two sisters to be named Florida. False anecdotal evidence aside (George Foreman named them "George A," "George B," etc.), that is not realistic. He dismissed the effect, saying that if f<<1, then f^2 was negligible. The problem with that was that every term used in his solution used f^1 or f^2.
To finish off Mlodinow, having a rare girl's name in a two-child family actually increases the probability if two girls. You've argued this increase elsewhere, although you use it justify decreasing the probability. Go figure. But the answer to the Florida question, if you accept his approach, is "a little more than 1/2."
And finally, Martin Gardner retracted the 1/3 solution you provide here. He said it needs more information, and gave examples. But it doesn't - without explicit statements that support 1/3. The answer has to be 1/2.
If "it’s reasonable to assume that all families with a girl are equally likely" were true, than if you tell us about a boy "it’s reasonable to assume that all families with a boy are equally likely" is just as true. Which means that the only reasonable assumption with a BG family is that you tell us about both the boy and the girl. There really is no way around this paradox: you are assuming two different things about the same family.
I have a question about the use of 'at least'. If we look at the Mr. Smith problem, suppose we leave out those words 'at least'.
Mr. Smith has two children. One of them is a boy. What is the probability that both children are boys?
Now I might ask 'which one is a boy?' You should be able to tell me, right? Maybe you don't want to tell me, but is that the reason why the answer is not 1/2?
If you tell me which one is boy, I know that a specific child is a boy. That makes the sex of the other child indepedent. So the answer would be 1/2.
If you don't tell me which one is boy, I know that you know a specific child to be boy. The other child's sex would still be independent and the answer would still be 1/2.
In the original Mr. Smith problem (with 'at least'), it doesn't make sense at all to ask: 'which one?'
Isn't it better or required to always add the words 'at least' to these kind of problems, including the Florida riddle?
This is very interesting, but how do we know that unusual names are restricted to children of the same sex? E.g., my first child might be a boy named Jack, so I'm ready to take a big swing by the time my daughter is born.
The Florida problem did come from Mlodinow's book, but it was more like this: You recall that a distant relative has two children. You also recall that one is a one is a girl with an unusual name, like "Florida." But you don't recall of both are girls. What is the probability that both are girls?
There are several problems with Mlodinow's answer. Which was "A little less than 1/2" in the book, and (2-f)/(4-f) in reactions to the book where f is the frequency of the name "Florida." First, the original Two Child Problem was two problems: one with "The elder is a boy" and one with "At least one is a boy." The difference is that the first identified a specific child, so the answer was 1/2. Well, Mlodinow identified a specific child, didn't he?
And if you don't think he did identify one, note that the "elder child" establishes an order. The 1/2 answer applies with any kind of order. Like putting whichever child was more memorable to Leonard Mlodinow first.
Mlodinow also allowed two sisters to be named Florida. False anecdotal evidence aside (George Foreman named them "George A," "George B," etc.), that is not realistic. He dismissed the effect, saying that if f<<1, then f^2 was negligible. The problem with that was that every term used in his solution used f^1 or f^2.
To finish off Mlodinow, having a rare girl's name in a two-child family actually increases the probability if two girls. You've argued this increase elsewhere, although you use it justify decreasing the probability. Go figure. But the answer to the Florida question, if you accept his approach, is "a little more than 1/2."
And finally, Martin Gardner retracted the 1/3 solution you provide here. He said it needs more information, and gave examples. But it doesn't - without explicit statements that support 1/3. The answer has to be 1/2.
If "it’s reasonable to assume that all families with a girl are equally likely" were true, than if you tell us about a boy "it’s reasonable to assume that all families with a boy are equally likely" is just as true. Which means that the only reasonable assumption with a BG family is that you tell us about both the boy and the girl. There really is no way around this paradox: you are assuming two different things about the same family.
I have a question about the use of 'at least'. If we look at the Mr. Smith problem, suppose we leave out those words 'at least'.
Mr. Smith has two children. One of them is a boy. What is the probability that both children are boys?
Now I might ask 'which one is a boy?' You should be able to tell me, right? Maybe you don't want to tell me, but is that the reason why the answer is not 1/2?
If you tell me which one is boy, I know that a specific child is a boy. That makes the sex of the other child indepedent. So the answer would be 1/2.
If you don't tell me which one is boy, I know that you know a specific child to be boy. The other child's sex would still be independent and the answer would still be 1/2.
In the original Mr. Smith problem (with 'at least'), it doesn't make sense at all to ask: 'which one?'
Isn't it better or required to always add the words 'at least' to these kind of problems, including the Florida riddle?
This is very interesting, but how do we know that unusual names are restricted to children of the same sex? E.g., my first child might be a boy named Jack, so I'm ready to take a big swing by the time my daughter is born.