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The answer is indeed half. Many of the solutions give nice formulas to prove this, but there's a much simpler answer: it's half no matter what stopping conditions are chosen, because of the simple fact that each child born has a 50% chance of being a boy.
- akprasad March 19, 2016Don't believe it's that simple? Consider a random bit string (this is from random.org):
1001101001001100111010110011000111110000
What's the expected fraction of zeros? Half. Now introduce breaks after each one (boy born):
1 0011 01 001 0011 00111 01 011 0011 00011111 0000
Each string is a family that waited til their first boy. *Now* what's the expected fraction of zeros?
It didn't change.