Suppose that China’s population policy is modified so that every family could have children until either a boy is born or until a maximum of four children are born, whichever comes first. Assume that every family chooses to have as many children as possible under this policy, that multiple births are not figured in, and that boys and girls are equally likely at birth. What would be the average number of children per family for China? Describe how you arrived at your conclusion. (Hint: Start with 10,000 families and follow through to see how many total children there would be after all the children are born.)
To determine the average number of children per family under the modified policy in China, let's go through the scenario step by step.
1. Start with 10,000 families.
2. In each family, the process continues until either a boy is born or until four children are born, whichever comes first.
3. Let's analyze the possible outcomes for each family:
a. First child: The probability of having a boy or a girl is equal, so there is a 50% chance of having a boy as the first child and a 50% chance of having a girl.
- If a boy is born as the first child, the family stops having children, so only one child is born in this case.
- If a girl is born as the first child, the family tries to have another child.
b. Second child: Assuming the first child is a girl, the family now has another chance to have a boy or a girl.
- There is still a 50% chance of having a boy and a 50% chance of having a girl.
- If a boy is born as the second child, the family stops having children, so two children are born in this case.
- If a girl is born as the second child, the family tries for the third child.
c. Third child: Assuming the first two children are girls, the family continues trying to have a boy or a girl.
- The same 50% chance applies for each gender.
- If a boy is born as the third child, the family stops having children, so three children are born in this case.
- If a girl is born as the third child, the family tries for the fourth child.
d. Fourth child: Assuming the first three children are girls, the family attempts to have another child.
- Again, there is a 50% chance of having a boy and a 50% chance of having a girl.
- If a boy is born as the fourth child, the family stops having children, so four children are born in this case.
- If a girl is born as the fourth child, the family is not allowed to have another child, so only four children are born in this case.
4. Now, let's calculate the expected number of children per family:
- The probability of having one child is 50% (boy) or 50% (girl) = 1 child x 0.5 + 0 children x 0.5 = 0.5 child.
- The probability of having two children is 50% (girl, girl) or 50% (girl, boy) = 2 children x 0.5 + 0 children x 0.5 = 1 child.
- The probability of having three children is 50% (girl, girl, girl) or 50% (girl, girl, boy) = 3 children x 0.5 + 0 children x 0.5 = 1.5 children.
- The probability of having four children is 50% (girl, girl, girl, boy) or 50% (girl, girl, girl, girl) = 4 children x 0.5 + 0 children x 0.5 = 2 children.
5. To find the average number of children per family, we need to calculate the weighted average:
average = (probability of each outcome) x (number of children in that outcome)
average = (0.5 child) x (1 child) + (0.5 child) x (2 children) + (0.5 child) x (1.5 children) + (0.5 child) x (2 children)
average = 0.5 + 1 + 0.75 + 1 = 3.25 children
Therefore, the average number of children per family under the modified policy in China would be 3.25 children.