Problem 7 - Calculus for Biology and Medicine

Question

A family has two children. The younger one is a girl. Find the probability that the other child is a girl as well.

Step-by-Step Solution

Answer

The probability is \( \frac{1}{2} \).

Step 1: Define the Sample Space

In a family with two children, each child can be a boy (B) or a girl (G). Since the younger child is known to be a girl, we only consider the cases where the younger child is G. So, the sample space is {BG, GG} where the first letter represents the older child and the second the younger.

Step 2: Determine Favorable Outcomes

We are looking for the probability that the other child (the older one) is also a girl, which corresponds to the outcome GG in the sample space.

Step 3: Calculate the Probability

The probability that the other child is a girl is the number of favorable outcomes (GG) divided by the total number of possible outcomes in the sample space. There is 1 favorable outcome (GG) and 2 possible outcomes (BG, GG). Thus, the probability is \( \frac{1}{2} \).

Key Concepts

Sample Space

In probability, understanding the concept of a sample space is fundamental. A sample space includes all possible outcomes of a given situation or experiment. For instance, in the case of a family having two children, where we know the younger one is a girl, the possible combinations of the children's genders form the sample space.

Given that one child (the younger) is guaranteed to be a girl (G), each scenario within the sample space must account for this. Therefore, the potential outcomes we consider are:

• BG - where the older child is a Boy and the younger is a Girl.
• GG - where both the older and the younger children are Girls.

This complete list of outcomes—{BG, GG}—constitutes the sample space. Recognizing the sample space is a crucial first step in solving probability questions as it lays out all scenarios we need to consider.

Favorable Outcomes

Once the sample space is identified, the next step is to determine the favorable outcomes. Favorable outcomes are the scenarios within the sample space that satisfy the condition of the probability problem we are trying to solve.

In our example, the goal is to find out if the other child, specifically the older one, is also a girl. Thus, the favorable outcome is the scenario where both children are girls. From our sample space {BG, GG}, the favorable outcome is:

• GG - where both the older and younger children are Girls.

Understanding which outcomes are favorable is key to resolving the probability problem efficiently.

Probability Calculation

The final step in resolving this probability problem involves calculating the probability itself. This is done by using a simple formula: dividing the number of favorable outcomes by the total number of possible outcomes in the sample space.

With our scenario:

• Total possible outcomes: 2 (BG, GG)
• Favorable outcomes: 1 (GG)

The probability is therefore calculated as:\[\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{1}{2}\] This means there is a 50% chance, or a probability of 0.5, that the other child—older child in this case—is also a girl.
Accurately calculating probability helps us understand the likelihood of different outcomes based on the given conditions.