In this hypothetical society, every couple continues to have children until they have a girl, and then they stop. Each child is independently a girl with probability \( \frac{1}{2} \).

Question:

What fraction of the population will be girls in the long run?

Intuitive Trap

At first glance, one might think there will be more girls than boys, since each family stops only after a girl is born. But let’s explore this rigorously.

Mathematical Analysis

Let’s suppose there are \( N \) couples.

Births Per Family

Each family continues until a girl is born. The number of births per family follows a geometric distribution with success probability \( p = \frac{1}{2} \). The expected number of children per family is:

\[ \mathbb{E}[\text{children per family}] = \sum_{k=1}^{\infty} k \cdot \left(\frac{1}{2}\right)^k = 2 \]

So, for \( N \) families, there are \( 2N \) expected total children.

Gender Distribution

Since each child is independently a girl with probability \( \frac{1}{2} \), the gender ratio among all births remains 1:1.

Hence, the expected number of girls is:

\[ \mathbb{E}[\text{girls}] = N \]

And the expected number of boys is also:

\[ \mathbb{E}[\text{boys}] = N \]

Final Fraction

The fraction of the population that is girls is:

\[ \frac{\text{girls}}{\text{girls} + \text{boys}} = \frac{N}{N + N} = \boxed{\frac{1}{2}} \]

Conclusion

Despite the stopping rule that every family stops at the first girl, the fraction of girls in the population remains exactly 50%. The key insight is that although every family guarantees a girl, the number of boys before that girl balances the overall ratio.

Reference