BrainTeaser BasicProbability Bijection Intermediate

Problem - 4689

$\textbf{Boys v.s. Girls}$

In a remote town, people generally prefer boys over girls. Therefore, every married couple will continue giving birth to a baby until they have a son. Assuming there is fifty-fifty chance for a couple to give birth to a boy or a girl, what is the ratio of boys to girls in this town over many years?


$\textbf{Answer}$

$1:1$.

$\textbf{Analysis}$

This result can be intuitively understood by applying the principle of symmetry. Imaging that there is a neighboring town where baby girl is preferred. A couple will continue reproducing until they have a daughter. The girls-to-boys ratio in this town should be exactly the same as the boys-to-girls ratio in the original town because their situations are exactly symmetric. It follows that the ratio B:G should equal G:B where B stands for the number of boys and G stands for the number of girls. This implies B=G, or their ratio is $1:1$.

$\textbf{Note}$

In fact, this problem is essentially the same as # 4650.

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