The crux of the issue is much better illustrated by the simplified problem where if a man tells you one of his kids is a boy, the other child is a boy in 1/3 ca

The crux of the issue is much better illustrated by the simplified problem where if a man tells you one of his kids is a boy, the other child is a boy in 1/3 cases. That's intuitive looking at the possibilities: BG GB BB Where this analysis gets confusing is when you say that if you meet the man with one of his kids at random and it's a boy, the probability changes to 1/2. The reason is in that case one of the possible outcomes was indeed GG, even if the child you met was a boy. The difference is how much knowledge you had about the situation prior to making your measurement of the outcome. If that doesn't make your head spin even a little you're not really human. I was able to make peace with these problems when I read about the Two Envelopes Paradox and realized that it's not really possible to pick a random integer, because by picking a random integer you're essentially limiting the domain of integers you've picked from to a finite one. If you can wrap your head around that it might help you save your sanity.