"At least one girl" versus "a girl answers the door"
A family has two children, each independently equally likely to be a boy or a girl. Consider two scenarios:
- You are told, as a fact about the family, that at least one child is a girl.
- You knock on the door and a girl answers; assume that if the family has one child of each sex, either is equally likely to come to the door.
In each scenario, what is the probability that both children are girls?
Show a hint
In scenario 1 you learn a property of the whole family. In scenario 2 you meet one specific child. Do those carry the same information?
Your answer
This one is open-ended. Work it through, then check your reasoning against the full solution.