Question

In \(1978,\) in an effort to reduce population growth, China instituted a policy that allows only one child per family. One unintended consequence has been that, because of a cultural bias toward sons, China now has many more young boys than girls. To solve this problem, some people have suggested replacing the one-child policy with a one-son policy: A family may have children until a boy is born. Suppose that the one-son policy were implemented and that natural birth rates remained the same (half boys and half girls). Using geometric series, compare the total number of children under the two policies.

Short answer

Answer: Under the one-son policy, the expected number of children born per family is twice as many as under the one-child policy.

Step-by-step solution

Calculate the expected number of children per family under the one-child policy.

Under the one-child policy, each family has one child, regardless of the gender. Therefore, the total number of children per family will always be 1.

Analyze the one-son policy using geometric series.

Under the one-son policy, a family stops having children once they have a boy. The series for the number of children born depends on the probability of having a boy. We can represent this process using a geometric series, with the number of children born representing the terms of the series. The first term represents the probability of having a boy in the first child, which is 1/2. The second term represents the probability of not having a boy in the first child (i.e., having a girl) and then having a boy in the second child, which is (1/2)*(1/2) = 1/4. The third term represents the probability of having a girl in both the first and second child and then having a boy in the third child, which is (1/2)*(1/2)*(1/2) = 1/8, and so on. Let's calculate the expected number of children per family under the one-son policy using this geometric series.

Calculate the sum of the geometric series for the one-son policy.

The sum of the geometric series representing the process of having children under the one-son policy can be calculated using the formula for the sum of an infinite geometric series: \(S = \frac{a}{1-r}\) where \(a\) is the first term of the series, \(r\) is the common ratio (in this case, 1/2), and \(S\) is the sum of the series. \(S = \frac{1/2}{1-1/2} = \frac{1/2}{1/2} = 1\)

Calculate the expected number of children per family under the one-son policy.

To find the expected number of children per family under the one-son policy, we need to multiply the probability of each term in the series by the number of children represented by that term and add these products up. Term 1: The probability of having a boy on the first try is 1/2, so this term contributes (1/2) * 1 = 1/2 child. Term 2: The probability of having a girl and then a boy is 1/4, so this term contributes (1/4) * 2 = 1/2 child. Term 3: The probability of having two girls and then a boy is 1/8, so this term contributes (1/8) * 3 = 3/8 child. Adding up these terms for all subsequent children, we get the infinite series: Expected number of children per family under the one-son policy = 1/2 + 1/2 + 3/8 + ... Using the same formula for the sum of an infinite geometric series as before, we get: \(S = \frac{1/2}{1-1/2} = \frac{1/2}{1/2} = 2\)

Compare the total number of children under the two policies.

Under the one-child policy, each family has only 1 child. Under the one-son policy, the expected number of children per family is 2. Comparing these two results, we can see that the one-son policy would result in twice as many children being born compared to the one-child policy.