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The way the puzzle is posed is ambiguous, as it does not state exactly how the trial is carried on. It may be reworded in (to my understanding) 2 different reasonnable ways, leading to 2 different answers.
1) Out of the set of 4 boys and 4 girls, that is 4 families representing the 4 combinations BB,BG,GB,GG, you choose at random a boy (rejecting the trial if you get a girl). Then the probability of the other member of the family being a boy is 1/2. This is the same as Ariskatsaris scenario B) below.
2) Out of the 4 families, you chose at random one that has at least one boy (rejecting the family with 2 girls), then the probabilty is is 1/3.
The way the puzzle is worded appears to me closer to scenario #2 than #1, but this my biased interpretation.