Problem 24 - College Algebra

Question

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{\mathrm{MMM},\) MMF. MFM. MFF. FMM. FMF, FFM, FFF . Find the probability of selecting a family with at least two female children.

Step-by-Step Solution

Answer

The probability of selecting a family with at least two female children is 0.5 or 50%.

Step 1: Identify the favorable outcomes

The first step is to define the favorable outcomes - these are the combinations which include at least two female children. The combinations that fit this criterion are: FFM, FMF, MFF, and FFF. This gives a total of 4 favorable outcomes.

Step 2: Define the total number of outcomes

The total sample space for a family with three children, where each child could either be male or female is defined in the problem as: {MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF}. There is a total of 8 outcomes.

Step 3: Calculate the Probability

The probability is calculated by taking the ratio of the number of favorable outcomes to the total number of outcomes. Hence, the probability of selecting a family with at least two female children is \(\frac{4}{8} = 0.5\) or 50%.

Key Concepts

Sample Space

In probability, the term "sample space" refers to the set of all possible outcomes that can occur in a given experiment or scenario. In the context of the problem at hand, we are examining a scenario where a family has three children. Each child could either be male, denoted as \(M\), or female, denoted as \(F\). Therefore, each potential combination of gender outcomes for these three children constitutes the "sample space."

For this problem, the sample space is presented as \( \{\text{MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF}\} \). Each outcome lists a possible configuration of male and female children, and with three children each having two gender possibilities, there are indeed \(2^3 = 8\) potential outcomes.

Understanding the sample space is crucial as it lays the foundation for determining probabilities in similar statistical problems. Every event's probability is calculated with respect to these possible outcomes, allowing us to identify and count specific conditions or "favorable outcomes."

Favorable Outcomes

"Favorable outcomes" in probability refer to the specific outcomes within the sample space that meet a given criterion we are interested in. For our problem, the favorable outcomes are those scenarios where the family has at least two female children.

Looking at our defined sample space \( \{\text{MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF}\} \), we identify the favorable outcomes as \(\text{MFF, FMF, FFM, and FFF}\). Each of these combinations has two or more female children.

Counting the favorable outcomes gives us a total of four possible events. Recognizing these scenarios relies on clearly understanding the condition stated in the problem, which helps in efficiently narrowing down the relevant outcomes. Identifying favorable outcomes is a critical step to determining the probability of an event.

Ratio of Outcomes

The "ratio of outcomes" is the mathematical calculation that expresses probability as a fraction. It involves comparing the number of favorable outcomes to the total number of possible outcomes from the sample space.

When calculating the probability of the family having at least two female children, we determined there were 4 favorable scenarios out of a total of 8 possible outcomes. This gives a probability expressed as the ratio \(\frac{4}{8}\).

Simplifying this fraction gives us \(\frac{1}{2}\), which can also be expressed as 0.5 or 50%. This means there is a 50% chance of selecting a family with at least two female children. Ratio calculations like these are central to translating the principle of probability into practical, easily understood terms.