Problem 73 - Algebra and Trigonometry

Question

The "Second Son" Paradox Mrs. Smith says, "I have two children. The older one is named William." "Mrs. Jones replies, "One of my two children is also named William." For each woman, list the sample space for the genders of her children, and calculate the probability that her other child is also a son. Explain why these two probabilities are different.

Step-by-Step Solution

Answer

Mrs. Smith: 1/2, Mrs. Jones: 1/3. Information specificity affects probability.

Step 1: Understanding the Sample Space for Mrs. Smith

Since Mrs. Smith's older child is named William, we know that the order of her two children is boy-girl (BG), or boy-boy (BB). Therefore, the sample space for the genders of Mrs. Smith's children is \( \{ \text{BG}, \text{BB} \} \).

Step 2: Calculating Probability for Mrs. Smith

We are interested in the probability that Mrs. Smith's other child is also a boy. Given the sample space \( \{ \text{BG}, \text{BB} \} \), only the \( \text{BB} \) outcome satisfies this condition. Thus, the probability is \( \frac{1}{2} \).

Step 3: Understanding the Sample Space for Mrs. Jones

Mrs. Jones states that one of her two children is named William. Her sample space is \( \{ \text{BB}, \text{BG}, \text{GB} \} \), as this covers all combinations where at least one child is a boy.

Step 4: Calculating Probability for Mrs. Jones

We want the probability that the other child is a boy given there is one boy. This includes both \( \text{BB} \) and one part of \( \text{BG} \) and \( \text{GB} \). However, if we consider both children being boys, only the \( \text{BB} \) scenario fits fully. Out of all scenarios \( \{ \text{BB}, \text{BG}, \text{GB} \} \) and considering symmetry, the \( \text{BB} \) case has probability \( \frac{1}{3} \).

Step 5: Comparing the Probabilities

Mrs. Smith's probability of having another son is \( \frac{1}{2} \) because we know the order of children. Mrs. Jones' probability is \( \frac{1}{3} \) because we only know one of the children is a boy, without specifying order. These differ due to conditional vs. unordered information.

Key Concepts

Sample Space

When solving probability problems, the concept of a sample space is fundamental. It represents all possible outcomes of an experiment or situation. For Mrs. Smith, since it is given that her older child is named William, her sample space focuses solely on the possible gender combinations of her children, knowing the first is a boy. The options are either a boy-girl (BG) or boy-boy (BB). Thus, her sample space is \( \{ BG, BB \} \).

In contrast, Mrs. Jones' statement only mentions one child named William, without specifying if that child is the first or the second. This means her sample space includes all possible gender combinations where at least one child is a boy. Consequently, her sample space is \( \{ BB, BG, GB \} \), covering scenarios where the boy could be either the first or the second child.

Understanding these sample spaces helps set the stage for accurate probability calculations later.

Probability Calculation

Probability calculation relies on using the sample space to determine the likelihood of specific outcomes. For Mrs. Smith, given the sample space \( \{ BG, BB \} \), we calculate the probability of her other child also being a boy. Only one outcome in this space, \( BB \), satisfies this condition. Therefore, the probability is simply \( \frac{1}{2} \).

For Mrs. Jones, her sample space \( \{ BB, BG, GB \} \) involves a more diverse set of possibilities. She wants to assess the probability that the other child is a boy when one is already known to be a boy. Among the three scenarios, BB is the only outcome where both children are boys. Thus, her probability calculation yields \( \frac{1}{3} \). This shows how knowing more specifics, such as the order of children, can impact probability outcomes.

Gender Combinations

Gender combinations are an essential aspect of understanding our sample spaces and, by extension, our probability calculations. In problems like this, we encounter different combinations such as Boy-Boy (BB), Boy-Girl (BG), and Girl-Boy (GB), which we must consider based on given conditions.

For instance, Mrs. Smith is clear about which of her children is a boy. This limits her gender combinations to \( \{BB, BG\} \), excluding the possibility of \( GB \).

Meanwhile, Mrs. Jones offers a less specific clue, only mentioning one child, leaving multiple gender combinations open. This necessitates including \( BB, BG, \) and \( GB \), acknowledging any potential position for the son. This aspect significantly influences the probabilities of different outcomes.

Conditional Probability

Conditional probability examines how the probability of an event can change when additional information is introduced. In this exercise, the changes stem from each mother's different revelations about their children.

For Mrs. Smith, where the older child is definitively a boy, the probability becomes conditional upon knowing one child's definite gender and position, making it \( \frac{1}{2} \) that the younger child is also a boy in her specific sample space.

However, Mrs. Jones's less specific statement creates a different condition: one randomly chosen child is a boy. This affects the overall probability, making it necessary to calculate based on a broader set (\(\{BB, BG, GB\} \)), reducing the probability to \(\frac{1}{3}\) that the other child is also a boy. By understanding these shifting probabilities, we gain insight into how conditions and available information directly influence probability outcomes.