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Chapter 4: Problem 19

A classical probability problem involves a king who wanted to increase the proportion of women by decreeing that after a mother gives birth to a son, she is prohibited from having any more children. The king reasons that some families will have just one boy, whereas other families will have a few girls and one boy, so the proportion of girls will be increased. Is his reasoning correct? Will the proportion of girls increase?

Short Answer

Expert verified
No, the proportion of girls remains 50%.

Step by step solution

01

- Understand the Problem

The king implements a rule where mothers stop having children after giving birth to a son. The expectation is that this rule will increase the proportion of girls in the population.
02

- Define Probabilities

Let's define the probability of having a boy as P(B) = 0.5 and the probability of having a girl as P(G) = 0.5. These probabilities reflect that there is an equal chance of having a boy or a girl.
03

- Consider Number of Births in Families

Each family will keep having children until they have a boy. Some families may have one child (a boy), some may have multiple girls followed by a boy. Each child's birth is an independent event with probabilities P(B) and P(G) both equal to 0.5.
04

- Calculate Expected Number of Children Per Family

The number of children a family will have depends on the number of girls before the first boy. For example, if a family has 1 girl and then a boy, they have 2 children. Expected number of children until a boy appears is represented by a geometric distribution. The expected value E(X) of a geometric distribution with success probability P(B) = 0.5 is 1 / 0.5 = 2.
05

- Determine Expected Number of Girls

In each family, the expected number of girls is 1. It is calculated by taking the probability P(G)=0.5 for each child and summing it over the expected number of births E(X)=2. Therefore, 0.5 * 2 = 1 girl per family.
06

- Determine Expected Proportion

Each family is expected to have 1 girl and 1 boy. Thus, the proportion of girls (girls / total children) will be 1 / (1 + 1) = 0.5 or 50%, which is the same as before the king's rule.
07

- Conclusion

Since the expected number of girls remains 1 per family and 1 boy per family, the proportion remains 50%. The king鈥檚 reasoning is incorrect; the proportion of girls will not increase as a result of his decree.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

geometric distribution
In probability theory, a geometric distribution is a discrete distribution that models the number of trials needed for a success in repeated, independent Bernoulli trials (trials with two possible results: success or failure). In this problem, each birth is considered a trial with two outcomes: a boy (success) or a girl (failure). The probability of success, having a boy, is P(B) = 0.5 and similarly, the probability of a girl is P(G) = 0.5.

This distribution helps us understand the scenario where families keep having children until they have a boy. The number of children a family has follows a geometric distribution because the first success (boy) ends the process.
probability of birth outcomes
Probability is the measure of the likelihood that an event will occur. In this exercise, we are concerned with the probability of having boys and girls during childbirth. (P(B) = 0.5) represents the probability of having a boy, and (P(G) = 0.5) represents the probability of having a girl. These probabilities assume that each birth is independent of the previous ones.

Given these probabilities, each child鈥檚 birth does not affect the upcoming births, meaning that the probability remains the constant 0.5 for every single child born, irrespective of the previous birth outcomes.
expected value
Expected value is a core concept in probability theory that provides a measure of the center of a probability distribution. In the context of this exercise, the expected value helps us determine the average number of girls and boys each family will have under the king's rule.

To find the expected number of children until a boy appears, we use the properties of geometric distribution with P(B) = 0.5. The expected value (E(X)) of a geometric distribution is calculated as 1/P(B). Thus, the expected number of children is E(X) = 1/0.5 = 2. This means that, on average, each family will have 2 children. Since each child has an equal probability of being a girl, the expected number of girls in each family is 0.5 times the expected number of children, which is 0.5 * 2 = 1 girl per family.

As a result, the expected number of boys per family is also 1. Hence, the proportion of girls in each family will be 1 girl to 1 boy, or 50%, implying that the king鈥檚 decree does not change the overall gender ratio in the population.

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Most popular questions from this chapter

Find the probability. If you make random guesses for 10 multiple choice SAT test questions (each with five possible answers), what is the probability of getting at least 1 correct? If these questions are part of a practice test and an instructor says that you must get at least one correct answer before continuing, is there a good chance you will continue?

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