This problem involves calculating the expected ratio of boys to girls in a country where each family continues to have children until they have a boy. Let’s break it down step by step:

Assumptions:

• Each child born is independently a boy or a girl with equal probability (50% chance for each).
• Families stop having children once they have a boy.

Breakdown of possibilities:

1. First child is a boy: This family has 1 child, which is a boy.

   • Probability = 1/2.
   • Number of boys = 1, Number of girls = 0.

2. First child is a girl, second is a boy: This family has 2 children, 1 girl and 1 boy.

   • Probability = (1/2) * (1/2) = 1/4.
   • Number of boys = 1, Number of girls = 1.

3. First two children are girls, third is a boy: This family has 3 children, 2 girls and 1 boy.

   • Probability = (1/2) * (1/2) * (1/2) = 1/8.
   • Number of boys = 1, Number of girls = 2.

4. First three children are girls, fourth is a boy: This family has 4 children, 3 girls and 1 boy.

   • Probability = (1/2) * (1/2) * (1/2) * (1/2) = 1/16.
   • Number of boys = 1, Number of girls = 3.

Generalizing the process:

For each family, the number of boys is always 1, while the number of girls depends on how many children are born before the boy arrives. If the family has $n-1$ girls, then they have $n$ children in total, with 1 boy and $n-1$ girls.

The expected number of girls can be calculated by considering all possible outcomes.

Expected number of girls per family:

The expected number of girls per family is the sum of the number of girls weighted by their probabilities:

$\text{Expected number of girls} = 0 \times \frac{1}{2} + 1 \times \frac{1}{4} + 2 \times \frac{1}{8} + 3 \times \frac{1}{16} + \dots$

This is a sum of an infinite geometric series:

$E(\text{girls}) = \sum_{n=1}^{\infty} \frac{n-1}{2^n}$

This sum can be simplified to $1$. So, the expected number of girls is 1 per family.

Ratio of boys to girls:

Since each family has exactly 1 boy and the expected number of girls is 1, the proportion of boys to girls in the country is:

$\frac{\text{Number of boys}}{\text{Number of girls}} = \frac{1}{1} = 1$

Conclusion:

The proportion of boys to girls in the country is 1:1.

Would you like further details on how this result is derived? Or do you have any additional questions on similar topics?

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Related questions:

1. How would the proportion change if families had a different stopping rule, like having a set number of children?
2. What would happen if the probability of having a boy or a girl were not equal?
3. Can you explain how the expected number of girls was calculated in more detail?
4. How does this problem relate to the "Gambler’s Ruin" problem in probability theory?
5. If families continued having children regardless of gender, how would the boy-to-girl ratio change?

Tip: When faced with probability problems involving sequences or events with fixed outcomes (like having children until a boy is born), drawing out the possibilities and using geometric series can often help simplify the problem.