A colleague, Eugene Salamin, came up with what I would consider the "Book" solution:

Phooey, this isn't at all a mathematical puzzle. A social convention cannot override biology, so the proportion of boys and girls is the biologically determined one, nominally 1/2, 1/2.

I didn't immediately understand his reasoning. But if all families are enumerated 1,2,3,... and you imagine each family's sequence of children placed in numerical order to make one infinite (or very long) sequence, then the resulting sequence of B's and G's is statistically identical to one you would get by repeatedly flipping a fair coin.

Viewed this way, the rule for stopping when the first B is reached is clearly a red herring! And clearly the proportion of boys and girls will be equal. (At least asymptotically, with probability 1, by the Strong Law of Large Numbers.)

(Likewise, if the original question is varied so that Prob(B) = p and Prob(G) = q, p+q=1, then by the same reasoning the ultimate proportions of boys and girls are p and q, respectively.)

P.S. On the other hand, this does not work for each possible stopping rule. Say we're back to the usual assumption of each birth having an equal chance of being a boy or girl. In an imaginary world, suppose each family stopped having children when the proportion of the girls in their family first exceeded 2/3 (which will occur eventually in each family with probability 1). Then the ratio of girls to boys in the population will clearly be greater than 2.

A colleague, Eugene Salamin, came up with what I would consider the "Book" solution:

Phooey, this isn't at all a mathematical puzzle. A social convention cannot override biology, so the proportion of boys and girls is the biologically determined one, nominally 1/2, 1/2.

I didn't immediately understand his reasoning. But if all families are enumerated 1,2,3,... and you imagine each family's sequence of children placed in numerical order to make one infinite (or very long) sequence, then the resulting sequence of B's and G's is statistically identical to one you would get by repeatedly flipping a fair coin.

Viewed this way, the rule for stopping when the first B is reached is clearly a red herring! And clearly the proportion of boys and girls will be equal. (At least asymptotically, with probability 1, by the Strong Law of Large Numbers.)

(Likewise, if the original question is varied so that Prob(B) = p and Prob(G) = q, p+q=1, then by the same reasoning the ultimate proportions of boys and girls are p and q, respectively.)

P.S. On the other hand, this does not work for each possible stopping rule. Say we're back to the usual assumption of each birth having an equal chance of being a boy or girl. In an imaginary world, suppose each family stopped having children when the proportion of the girls in their family first exceeded 2/3. Then the ratio of girls to boys in the population will clearly be greater than 2.

A colleague, Eugene Salamin, came up with what I would consider the "Book" solution:

Phooey, this isn't at all a mathematical puzzle. A social convention cannot override biology, so the proportion of boys and girls is the biologically determined one, nominally 1/2, 1/2.

I didn't immediately understand his reasoning. But if all families are enumerated 1,2,3,... and you imagine each family's sequence of children placed in numerical order to make one infinite (or very long) sequence, then the resulting sequence of B's and G's is statistically identical to one you would get by repeatedly flipping a fair coin.

Viewed this way, the rule for stopping when the first B is reached is clearly a red herring! And clearly the proportion of boys and girls will be equal. (At least asymptotically, with probability 1, by the Strong Law of Large Numbers.)

(Likewise, if the original question is varied so that Prob(B) = p and Prob(G) = q, p+q=1, then by the same reasoning the ultimate proportions of boys and girls are p and q, respectively.)

A colleague, Eugene Salamin, came up with what I would consider the "Book" solution:

Phooey, this isn't at all a mathematical puzzle. A social convention cannot override biology, so the proportion of boys and girls is the biologically determined one, nominally 1/2, 1/2.

I didn't immediately understand his reasoning. But if all families are enumerated 1,2,3,... and you imagine each family's sequence of children placed in numerical order to make one infinite (or very long) sequence, then the resulting sequence of B's and G's is statistically identical to one you would get by repeatedly flipping a fair coin.

Viewed this way, the rule for stopping when the first B is reached is clearly a red herring! And clearly the proportion of boys and girls will be equal. (At least asymptotically, with probability 1, by the Strong Law of Large Numbers.)

(Likewise, if the original question is varied so that Prob(B) = p and Prob(G) = q, p+q=1, then by the same reasoning the ultimate proportions of boys and girls are p and q, respectively.)

P.S. On the other hand, this does not work for each possible stopping rule. Say we're back to the usual assumption of each birth having an equal chance of being a boy or girl. In an imaginary world, suppose each family stopped having children when the proportion of the girls in their family first exceeded 2/3 (which will occur eventually in each family with probability 1). Then the ratio of girls to boys in the population will clearly be greater than 2.