Thanks for all the replies to my original post. With your help, and my own counter productive thoughts while at work, I’ve come to understand the answer is 2/3.
The idea that the smallest information changes the probability is quite annoying to the real world brain, the phrasing is just right to provoke a quick (wrong) answer.
Adding any identifying aspect to the female child makes it 50%. It doesn’t change the real world chance of either child, just the theoretical probability we can expect from the information we have.
The only thing you changed was “my other child” to “Claire’s sibling”. The two are synonymous, so that change cannot affect the logic of the question in any way. You’re just interpreting the same question two different ways.
Of course, your first way is wrong.
Here’s the confusion: birth order doesn’t matter any more, because what we’re specifying about the child is not whether she’s older or younger but rather what she’s doing tomorrow. So the full universe of possibilities is as follows:
Partying girl, no-party girl
Partying girl, no-party boy
Partying boy, no-party girl
Partying boy, no-party boy
Each is equally likely, and we’ve eliminated 3 and 4, leaving just 1 and 2. So the answer is 1/2.
(Note that if we assume that /both/ kids could be going to a slumber party, then your answer of 2/3 is correct. But then it’s just as correct for the second question as well.)
Powers &8^]
No, “my other child” refers to one of two children in a family. The units being discussed are families, and the specific universe of families being discussed is those with two children, one of whom is a girl.
“Claire’s sibling” refers to a child who is the sibling of a girl. The units being discussed are girls, and the specific universe of girls being discussed is those with one sibling.
Yes, I’m interpreting the question two different ways. That was precisely my point. But neither interpretation is wrong.
The coworker says, “I have two kids.” Later he says, “My daughter Claire is going to a slumber party this weekend.” And then someone asks you, “Hey, coworker mentioned a daughter; what are the odds that his other kid is a son?”
One way to phrase the question is, “In my coworker’s family, which has two kids, one is a girl. What are the odds the other is a boy?”
So, we look at all the families with two kids (four kinds):
Girl A and Boy A
Boy B and Girl B
Girl C and Girl D
Boy C and Boy D
We eliminate the families with no girls (one kind):
Girl A and Boy A
Boy B and Girl B
Girl C and Girl D
[del]Boy C and Boy D[/del]
We look at the remaining families with boys (2 kinds) Girl A and Boy A
Boy B and Girl B
Girl C and Girl D
[del]Boy C and Boy D[/del]
And without (1 kind) Girl A and Boy A
Boy B and Girl B Girl C and Girl D
[del]Boy C and Boy D[/del]
And we figure the odds that this family with a girl also has a boy: 2/3. And we completely ignore the fact that the daughter is named Claire, or that she even has a name, because we’re not talking about girls, we’re still talking about families. The only reason her name was even of interest is that it accompanied the fact that one of the family’s two children is a girl.
Another way to interpret it is, “My coworker has a daughter, Claire. He has two kids, so therefore, she has a sibling. What are the odds her sibling is a boy?”
So, we look at all the girls with siblings (4 kinds)
Girl A and Boy A
Boy B and Girl B
Girl C and Girl D
We count those with boy siblings (2 kinds) Girl A and Boy A
Boy B and Girl B
Girl C and Girl D
And those without (2 kinds) Girl A and Boy A
Boy B and Girl B Girl C and Girl D
And we figure the odds that this girl Claire has a brother: 1/2. And the only reason her name or her plans or any of those other details are relevant is because now we’re not talking about families, we’re talking about girls like Claire.
I don’t see how you can say anything about one of the children and not change the universe. You have changed the statement from “one of my children is a girl” to “this child is a girl”. Which child? Claire, or the one going to a sleepover, or the one wearing pigtails, or whatever additional information was provided.
But I agree with your explanation of the outcomes for the different perspectives.
Okay, I see now… but what you’re missing is that it’s not the wording of the sibling-referrent (“my other child” versus “Claire’s sibling”) that limits the universe and results in the 1/2 probability; it’s the identification of Claire. The sibling and how you refer to him/her has nothing to do with it; it’s entirely about Claire.
You’re getting the right result in the second case, but the wrong one in the first case because you’re paying attention to the wrong thing. In the first case, the universe of families being discussed is not those with two children, one of whom is a girl, it’s those with two children, a girl named Claire and another one. That universe is identical to the one of girls named Claire with one sibling.
Powers &8^]
Here is the problem in a nutshell. By stating “other kid” in that question, you have to look at what is meant by “other”. In this case, other than the specified kid, the daughter, Claire, who is going to a slumber party. Whether you know her name is Claire, or only know she is going to a slumber party, you have specified a kid. So you can only look at the other kid, i.e. the “not-Claire” kid or the “not going to the sleepover” kid.