The Two Children Paradox

The problem
a family of two children??clue: at least one is a boyP(both boys) = ?

Someone tells you:

"I have exactly two children. At least one of them is a boy."

What is the probability that both of their children are boys?

Tempting (but wrong)
"the other child is 50 / 50"?known½ boy, ½ girl?⇒ P(both boys) = ½ ✗

The instinct: the other child is independent — fifty-fifty boy or girl. We're told one is a boy, so the chance both are boys equals the chance the other one is a boy, which is 12\tfrac{1}{2}.

This treats one specific child as known, then asks about the other in isolation. That's a different problem.