You're assuming a different selection model, in which a father of two boys is twice as likely to volunteer information if he has two boys: "Pick a child at rand

You're assuming a different selection model, in which a father of two boys is twice as likely to volunteer information if he has two boys: "Pick a child at random, then if it's a boy - say: I have a boy [blablabla] what is the other?" However, you're doing this on an already selected sample by discounting all the girl-girl pairs. The selection rule in your logic then becomes a two-step "If you have at least one boy, then pick a child at random, then if it's a boy, say: [blablabla]" Given that, your analysis is correct. But, seeing as it is a wildly different selection model than what everyone else seem to work with, you should be explicit about it.