Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the probability that a family with three children will have the given composition. The two oldest children are girls.

Step-by-Step Solution

Answer

The probability that a family with three children will have the given composition (i.e., the two oldest children are girls) is \(1\), or \(100%\).

Step 1: Determine the sample space for the youngest child's gender

The youngest child could be a boy or a girl. Therefore, the possible outcomes are: 1. Boy 2. Girl

Step 2: Calculate the probability of each outcome

Since there are two possible outcomes and we assume each has an equal chance of occurring, the probability of each outcome is: \( P(Boy) = \frac{1}{2} \) \( P(Girl) = \frac{1}{2} \)

Step 3: Identify the desired outcome

We are looking for the probability of the family having two girls as the oldest children. Since the oldest two children are already girls, we are actually looking for the probability of the third child, the youngest, being of any gender. Since the probability of the third child being a boy or a girl is equal, the probability of the given composition (two oldest children being girls) is already certain. Therefore the probability is: \( P(Given\ Composition) = 1\)

Step 4: Conclusion

The probability that a family with three children will have the given composition (i.e., the two oldest children are girls) is 1, or 100%.

Key Concepts

Understanding the Sample Space

The concept of sample space is fundamental in understanding probability, as it encompasses all the possible outcomes that could arise from a particular situation. In the context of a family with three children, the sample space for the gender of the youngest child consists of only two outcomes: the child being a girl or a boy. This can be visually represented as a simple list:

• Boy
• Girl

Knowing the sample space is crucial—it serves as the starting point for all probability calculations and provides a clear picture of what possibilities we're considering. If other characteristics, such as hair color or eye color, were also factors, the sample space would become much more complex, involving a longer list of possible outcomes.

The Mechanics of Probability Calculation

Once we understand the sample space, we can move on to probability calculation. Probability quantifies the likelihood of an outcome and is calculated by dividing the number of favorable outcomes by the total number of possible outcomes in the sample space. For example, if we're considering the gender of the youngest child in a three-children family, and the two oldest are already identified as girls, we're looking at a sample space with two equally likely outcomes:
\r

• Outcome 1 (Boy): The probability is \( P(Boy) = \frac{1}{2} \)
• Outcome 2 (Girl): The probability is \( P(Girl) = \frac{1}{2} \)

Each outcome has an equal probability because there's no reason to assume one sex occurs more frequently than the other at birth. The sum of these probabilities must equal 1, which is a fundamental rule in probability theory, representing the certainty that one of the outcomes in the sample space will occur.

Analyzing Outcomes in Probability

In discussing outcomes in probability, it's key to align our understanding of 'favorable' outcomes in the context of the problem at hand. In our exercise, we determine that the two oldest children in a family of three are girls. Therefore, the gender of the youngest child doesn't impact the event we're examining — the family already meets the specified composition regardless of whether the youngest child is a boy or a girl. Thus, the probability is certain or, numerically, \( P(Given\ Composition) = 1 \).\r
In a different scenario, if we were asked about the likelihood of having at least one boy in a three-children family, the analysis of outcomes would vary greatly. This extension into more complex probability questions demonstrates the importance of understanding the specifics of each probability scenario, and the need to evaluate how each outcome fits into the question posed.