The correct answer is 1/2. In statistical notation, he is asking us to calculate E[G]/E[B+G], the expected proportion of females in the total population. However, he turns it into the different question E[G/(B+G)], the expected proportion of females in an average family, which is not generally equal to the first expression (since families are of different sizes) and which in this case gives the incorrect answer of 30.6%. The guy is impervious to all the good arguments that have been posted to his blog pointing out his error.
His argument is exactly the same as if I headed down to the roulette tables in Vegas and placed bets on black, just making sure that at each session I stop when black hits. According to his "math", that strategy should provide a 69.4% win rate (slightly less once we account for 0 and 00, but still well above 50%). A sure-fire way to beat the house!
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