Question

A student claims in Problem 1.5 that if one child is a girl, the probability that both are girls is \(\frac{1}{2} .\) Use appropriate sample spaces to show what is wrong with the following argument: It doesn't matter whether the girl is the older child or the younger; in either case the probability is \(\frac{1}{2}\) that the other child is a girl.

Short answer

Given one child is a girl, the correct probability both are girls is \(\frac{1}{3}\), not \(\frac{1}{2}\).

Step-by-step solution

- Define the Sample Space

List all possible outcomes for the gender of two children. The possible outcomes are: BB (both boys), BG (older boy, younger girl), GB (older girl, younger boy), GG (both girls).

- Determine the Favorable Outcomes

Identify the favorable outcomes given that at least one child is a girl. This reduces our sample space to: BG, GB, and GG.

- Evaluate the Probability

Calculate the probability that both children are girls, given that at least one is a girl. The favorable outcome (both girls) is GG, and the total number of favorable outcomes is 3. Therefore, the probability is \(\frac{1}{3}\).

- Explain the Incorrect Argument

Explain why the student's argument is incorrect. The student's error is in stating that the probability is influenced by only one of the children's gender, ignoring the fact that the correct sample space must be considered, thus changing the probability to \(\frac{1}{3}\).

Key concepts

Understanding Sample Space

In probability theory, the sample space is a critical concept. It includes all possible outcomes of a particular experiment. For example, if we consider the genders of two children, each child can either be a boy or a girl. Thus, for two children, the sample space includes the following possible combinations:

• BB: Both children are boys
• BG: Older child is a boy and younger child is a girl
• GB: Older child is a girl and younger child is a boy
• GG: Both children are girls

This exhaustive list ensures that we consider every possible outcome, which is crucial for accurate probability calculations.

Exploring Conditional Probability

Conditional probability helps us understand the probability of an event, given that another event has occurred. It is denoted by \( P(A|B) \), indicating the probability of event A, given event B has occurred. To apply this, think about the problem where we know one child is a girl and want to find the probability that both children are girls. Our sample space is reduced to:

• BG
• GB
• GG

With this new sample space, the probability that both children are girls (GG) is \( \frac{1}{3} \), since there is 1 favorable outcome out of 3 possible outcomes. Therefore, considering given conditions is essential in obtaining accurate probabilities.

Probability in Gender Combinations

When we look at gender combinations, such as having two children, understanding the intricacies is vital. If we want to compute probabilities based on a known fact (e.g., one child is a girl), we must adjust our sample space accordingly. Originally, the combinations are:

• BB
• BG
• GB
• GG

However, knowing one child is a girl changes this to:

• BG
• GB
• GG

Now, we recalculate probabilities based on this adjusted sample space. For instance, the probability of having both girls (GG) becomes \( \frac{1}{3} \). This emphasizes the essential role of accurate sample spaces and conditions in determining the correct probabilities in gender combinations.