生育政策与男女比例 (Fertility Policy)¶

Problem¶

Question

English中文

If a family stops having children after giving birth to a boy, and continues to have children after giving birth to a girl until a boy is born, the question is: What will the gender ratio be in this place?

如果一个家庭生下男孩就不再生孩子，如果是女孩子就继续生，直到生出男孩为止，问：这个地方的男女比例会是多少？

Tip¶

Tip

English中文

There are many solutions, which can be approached from the perspectives of expectation, geometric distribution, probability, biology, etc.

很多种解法，可以从期望、几何分布、概率，生物学等角度入手。

Solutions¶

Solution1: Analysis¶

Solution1

English中文

Since there is only one boy in a family, asking about the ratio of boys to girls is equivalent to asking the average number of girls per family.

Let the number of girls in a family be \(X\), and the average number of girls per family be \(E(X)\), and the problem is transformed into solving \(1:E(X)\).

Since the probability of having a boy and a girl is equal, the distribution of X is as follows:

X	Sequence	Probability
0	Male	\(\frac{1}{2}\)
1	Female, Male	\((\frac{1}{2})^2\)
2	Female, Female, Male	\((\frac{1}{2})^3\)
3	Female, Female, Female, Male	\((\frac{1}{2})^4\)

According to the definition of expectation, \(E(X)\) satisfies the following equation:

\[ \begin{align} E(X) &= \frac{1}{2} \times 0 + (\frac{1}{2})^2 \times 1 + (\frac{1}{2})^3 \times 2 + \cdots \\ &= (\frac{1}{2})^2 \times 1 + (\frac{1}{2})^3 \times 2 + \cdots \end{align} \]

Therefore,

\[ 2E(X) = (\frac{1}{2})^1 \times 1 + (\frac{1}{2})^2 \times 2 + \cdots \]

Subtracting the two equations, we get:

\[ \begin{align} E(X) &= (\frac{1}{2})^1 \times 1 + (\frac{1}{2})^2 \times 1 + \cdots \\ &= lim_{n\to\infty} \frac{\frac{1}{2}(1-(\frac{1}{2})^n)}{1-\frac{1}{2}} \\ &= \frac{\frac{1}{2} \times 1}{\frac{1}{2}} \\ &= 1 \end{align} \]

因为一个家庭有且仅有一个男孩，所以问男女比例等价于问平均每个家庭女孩的数量。设家庭中女孩的数量为\(X\), 平均每个家庭女孩的数量则为\(E(X)\)，问题转化为求解\(1:E(X)\)。

由于生男孩和生女孩的概率相等，X的分布如下：

X	Sequence	Probability
0	男	\(\frac{1}{2}\)
1	女男	\((\frac{1}{2})^2\)
2	女女男	\((\frac{1}{2})^3\)
3	女女女男	\((\frac{1}{2})^4\)
...	...	...

根据期望的定义, \(E(X)\)满足下式：

\[ \begin{align} E(X) &= \frac{1}{2} \times 0 + (\frac{1}{2})^2 \times 1 + (\frac{1}{2})^3 \times 2 + \cdots \\ &= (\frac{1}{2})^2 \times 1 + (\frac{1}{2})^3 \times 2 + \cdots \end{align} \]

因此,

\[ 2E(X) = (\frac{1}{2})^1 \times 1 + (\frac{1}{2})^2 \times 2 + \cdots \]

两式相减得到：

\[ \begin{align} E(X) &= (\frac{1}{2})^1 \times 1 + (\frac{1}{2})^2 \times 1 + \cdots \\ &= lim_{n\to\infty} \frac{\frac{1}{2}(1-(\frac{1}{2})^n)}{1-\frac{1}{2}} \\ &= \frac{\frac{1}{2} \times 1}{\frac{1}{2}} \\ &= 1 \end{align} \]

Solution2: Analysis¶

Solution2

English中文

Define that when a family gives birth to a boy, the total number of children in the current family is \(X\), and the number of girls is \(Y\), then \(X\) meets the definition of geometric distribution (strictly speaking, it should be shifted geometric distribution), which is the probability distribution of the number of first successes in the Bernoulli test, and \(p=\frac{1}{2}\). According to the expected formula of geometric distribution, we have:

\[ E(X) = \frac{1}{p} = 2 \]

Because the number of boys is 1, the number of girls is \(Y=X-1\), so we have:

\[ E(Y) = E(X-1) = E(X) - 1 = 1 \]

That is, there is an average of 1 girl in each family, so the ratio of boys to girls is 1:1.

定义一个家庭生下男孩时，当前家庭孩子的总数量为\(X\), 女孩数量为\(Y\), 则\(X\)符合几何分布(严格来说应该是shifted geometric distribution)的定义，即为Bernoulli试验中第一次成功的次数的概率分布, 此时\(p=\frac{1}{2}\)。根据几何分布的期望公式，有:

\[ E(X) = \frac{1}{p} = 2 \]

因为男孩数量为1，所以女孩数量为\(Y=X-1\)，所以有:

\[ E(Y) = E(X-1) = E(X) - 1 = 1 \]

即每个家庭平均有1个女孩，所以男女比例为1:1。

Solution3: Elegant Analysis¶

Solution3

English中文

Consider a birth process as an event, let event A be the birth of a boy, and event B be the birth of a girl. Then: \(P(A) = \frac{1}{2}\), \(P(B) = \frac{1}{2}\). Regardless of the birth policy, it is ultimately a series of independent birth events, so the final male-female ratio is 1:1.

把一次生育过程看作一个事件，设事件A为生男孩，事件B为生女孩。则有: \(P(A) = \frac{1}{2}\), \(P(B) = \frac{1}{2}\). 无论生育政策如何，最终都是一系列相互独立的生育事件，所以最终的男女比例是1:1。

Solution4: Simulation¶

Solution4

English中文

We can simulate the birth process to obtain the numerical solution of the gender ratio under the current fertility policy:

import random

from enum import Enum


class Gender(int, Enum):
    MALE = 0
    FEMALE = 1


def birth_event() -> Gender:
    if random.randint(0, 1) == 0:
        return Gender.MALE
    else:
        return Gender.FEMALE


def family_event() -> int:
    female_count = 0
    while birth_event() != Gender.MALE:
        female_count += 1
    return female_count


def simulation(run_nums: int = 100000) -> None:
    male_count, female_count = 0, 0
    for _ in range(run_nums):
        female_count += family_event()
        male_count += 1
    print(f"Males: {male_count}, Females: {female_count}\n")
    print(f"Males/Female: {male_count / female_count}\n")


simulation()

Males: 100000, Females: 99963

Males/Female: 1.0003701369506717

我们可以通过模拟生育的过程来得到当前生育政策下性别比例的数值解:

import random

from enum import Enum


class Gender(int, Enum):
    MALE = 0
    FEMALE = 1


def birth_event() -> Gender:
    if random.randint(0, 1) == 0:
        return Gender.MALE
    else:
        return Gender.FEMALE


def family_event() -> int:
    female_count = 0
    while birth_event() != Gender.MALE:
        female_count += 1
    return female_count


def simulation(run_nums: int = 100000) -> None:
    male_count, female_count = 0, 0
    for _ in range(run_nums):
        female_count += family_event()
        male_count += 1
    print(f"Males: {male_count}, Females: {female_count}\n")
    print(f"Males/Female: {male_count / female_count}\n")


simulation()

Males: 100000, Females: 99519

Males/Female: 1.0048332479225073