全是女孩的世界？

All Girls World?

专题: Probability / 概率
难度: L2
来源: Brainstellar

题目详情

在一个“每家都想要女孩”的世界里，每个家庭会一直生孩子，直到生出女孩为止。

假设生男孩和生女孩概率相同（各 1/2），且没有其他性别。长期来看，男孩与女孩的比例会是多少？

In a world where everyone wants a girl child, each family continues having babies till they have a girl. What do you think will the boy-to-girl ratio be eventually?

Assuming probability of having a boy or a girl is the same and there is no other gender at the time of birth.

解析

最终比例仍是 $1:1$ 。

理由：每个家庭最终都会贡献 1 个女孩；而在得到这个女孩之前出生的男孩数量服从几何分布，期望为 1。把很多家庭加总后，男孩期望数量与女孩数量同阶，长期比例趋于 $1:1$ 。

Original Explanation

1:1

Solution

Suppose there are $N$ couples. First time, $N/2$ girls and $N/2$ boys are born. $N/2$ couples retire, and rest half try for another child.

Next time, $N/4$ couples give birth to $N/4$ girls and $N/4$ boys. Thus, even in the second iteration, the ratio is $1:1$ . It can now be seen that this ratio will always remain the same, no matter how many times people try to give birth to a favored gender.