妇女协会

Society of Women

专题: Probability / 概率
难度: L3
来源: OpenQuant

题目详情

你在雨林中冒险，突然接触到了一个与世隔绝的文明。在这个文明中，女孩是受优先考虑的，夫妇在生女孩之前不会停止生育孩子。假设一对夫妇生男孩和生女孩的可能性相等，那么多年后这个社会中女性的比例会是多少？

You are off venturing in a rainforest where you suddenly come into contact with an isolated civilization. In this civilization, girls are preferred and couples do not stop having children until they have had a girl. Assuming there is equal likelihood of a couple having a boy and a girl, what will the proportion of women in this society be after many years?

解析

假设这个文明中有 $n$ 夫妇。每对夫妇一旦生了女孩，就会停止生育。这意味着下一代将会存在 $n$ 的女孩。

如果女孩的分布呈 $p = \frac12$ 的几何分布，则每对夫妇平均会有 $2$ 个孩子，总共有 $2n$ 个孩子。这个社会中女孩的比例是： $\frac{n}{2n} = \boxed{\frac12}$

Original Explanation

Let's assume there are $n$ couples in this civilization. Each couple will stop having children once they have had a girl. This mean's there in the next generation there will exist exactly $n$ girls.

If the distribution of girls is geometric with $p = \frac12$ , each couple will on average have $2$ children, amounting to a total of $2n$ children. The proportion of girls in this society is:

\frac{n}{2n} = \boxed{\frac12}