paradox
Boy or girl paradox
Two children. At least one is a boy.
Two children. You hear that at least one is a boy. How often are both boys?
MoTuWeThFrSaSu
Two boys
One boy
Four families, equally likely. Lock a guess, then the ones ruled out leave.
Among the families that match, what share has two boys?
—Both boys, among families that match
—Families that match
—Long-run share
Two children give four equally likely families: boy-boy, boy-girl, girl-boy, and girl-girl.
The picture lines those four families up. The guess is the share with two boys among the families the news still allows.
Hearing that at least one is a boy removes only the girl-girl families. One of the three families left has two boys.
If the news is more specific, such as a boy born on Tuesday, fewer mixed families qualify, and the share with two boys moves from 1 in 3 toward 13 in 27.