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生到男孩为止

proportion of males to females

专题
Brainteaser / 脑筋急转弯
难度
L4

题目详情

某国准备可能的未来战争，传统只派男性参战，因此希望通过控制生育来提高人口中男性占比。

法律规定：每对已婚夫妇必须生孩子，并且必须一直生到生出一个男孩为止。

你预期该法律对人口男女构成会产生什么影响？

英文原题

A country is preparing for a possible future war. The country's tradition is to send only males into battle and so they want to increase the proportion of males to females in the population through regulating births. A law is passed that requires every married couple to have children and they must continue to have children until they have a male. What effect do you expect this law to have on the makeup of the population?

解析

在“每次生男/女概率均为 1/2 且相互独立”的假设下，总体男女比例不会改变，仍趋近于 1:1。

直观解释：每个家庭最终都会有且仅有 1 个男孩（最后一次出生），而女孩的数量是“在第一次男孩之前出现的女孩数”，其分布是几何分布的“失败次数”。

设每个家庭女孩数为 GG，男孩数为 B=1B=1。则

P(G=k)=(12)k+1,k=0,1,2,…\mathbb{P}(G=k)=\left(\frac12\right)^{k+1},\quad k=0,1,2,\ldots

于是

E[G]=∑k≥0k(12)k+1=1,\mathbb{E}[G]=\sum_{k\ge 0}k\left(\frac12\right)^{k+1}=1,

因此每个家庭期望女孩数等于 1，与男孩数相同，整体上男女比例不变。

英文解析

Under the assumption that the probability of having a boy or a girl is 1/21/2 and these events are mutually independent, the overall gender ratio remains unchanged and still approaches 1:11:1.

Intuitive explanation: Every family will eventually have exactly one boy (the last child born), while the number of girls is the "number of failures before the first success," which follows a geometric distribution.

Let GG be the number of girls per family and B=1B=1 be the number of boys per family. Then

P(G=k)=(12)k+1,k=0,1,2,…\mathbb{P}(G=k)=\left(\frac12\right)^{k+1},\quad k=0,1,2,\ldots

Thus

E[G]=∑k≥0k(12)k+1=1,\mathbb{E}[G]=\sum_{k\ge 0}k\left(\frac12\right)^{k+1}=1,

so the expected number of girls per family equals 1, which is the same as the number of boys. Consequently, the overall gender ratio remains unchanged.