Sorry; you are definitely right about that. I was so focused on the mom with the T-shirt that I did not even think about being able to count the states with the girl, but yes, the answer is obviously 50%.
No, I don't think the label "picked up from a police station" affects the problem the way you are thinking. What I believe you are not accounting for is that there are twice as many mixed families in the population as there are pure girl families.
Take 1000 two-child families (so as to intuitively ignore fluctuations). Families 1-250 are girl-girl, 251-500 had a girl then a boy, and 501-750 had a boy then a girl. Any of those 750 families could have made the quote in my post, yet there are 500 families with boy-girl, and 250 with girl-girl.
Or another example that I think speaks more directly to your post: you see a woman with a T-shirt reading "Proud Mother of Two" next to a girl who is obviously her daughter. What is the probability of her other child being a boy? Again, since there are twice as many mixed families as pure girl families, the odds are 2/3.
Can you give me your reasoning? I think the only conclusions you can get from that quote is that the person has two children, and at least one of those is a girl. Do you disagree with that?
If I am correct about that, then it matches the conditions discussed in the article and the answer would be 2/3 for a boy and a girl.
I don't agree; Jeff was not giving a quote. Instead it is just the relevant information abstracted from whatever the person said. By choosing the quote you did, you have added more information to the problem (at least when reading it with conversational English).
I think this would be a better quote of what the person might have said:"Both of my kids are driving me crazy! Just yesterday I had to pick one of them up from the police station--I grounded her for a month!" Pulling out the information corresponding to gender and family size would give only the information given in Jeff's post.
When applying math to the real world, you have to pull out the important information and deal with just that information. But here you are doing the opposite--trying to find a real world situation that applies to the math problem. In my opinion, your example does not quite apply.
(I don't know how the probabilities change when you account for hermaphrodites, but if it changes significantly enough so that approximately 66% is a bad answer, I would find that very interesting!)