I'm not sure I totally understand, but maybe someone can confirm this for me. If we add more "extraneous" information, it seems like it pushes the probability closer to (the naiive answer of) 1/2.
If we add lots of extraneous information, does it get really close? What if we did something like this:
I have two children, one of whom is a black-haired, blue-eyed son with an owl-shaped birthmark on his right leg born on a Tuesday in Argentina during an eclipse while a flock of 231 seagulls circled clockwise overhead.
What is the probability that I have two boys?
Am I just muddying the water, or is the probability vanishingly close to 2?
Forget about Thursday; someone tells you "I have two kids, at least one of which is a boy who is special". Now you get these cases for the two children: Bg, gB, Bb, bB, BB (where g is a girl, b is a boy, B is a special boy). Without the last case (two special boys), everything is symmetrical, and the probability that the second child is a boy is 1/2. The less likely it is that the guy has two special boys, the closer the answer is to 1/2.
This also means that the everyday answer is actually 1/3, because parents always think that all of their children are special, so the problem reverts to the simple Two Children Problem with the equally likely cases BG, GB, and BB (yes, the girls are special, too).
I have two children, one of whom is a black-haired, blue-eyed son with an owl-shaped birthmark on his right leg born on a Tuesday in Argentina during an eclipse while a flock of 231 seagulls circled clockwise overhead. What is the probability that I have two boys?
Am I just muddying the water, or is the probability vanishingly close to 2?