I.e. without "discarding", just giving some additional, but not complete, information on the random sample. Is adding information about the picked sample the same as discarding all contrarian samples? Why is this relevant?
It’s not a random family if it must have at least one girl. If you want to talk about a random family you can only make statements of the kind “one of the children is <gender>” where the gender depend on the specific family or “the family has between 0 and 2 girls”
of those 3 cases, 1/3 are both girls
If you phrase the question as “someone with two children tell you the gender of a random one, what is the chance the other is the same gender?” Chance is 50/50 because 50% will have BB or GG and the vaporizer isn’t active.