Problem 33 - A Graphical Approach to College Algebra

Question

Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2}\) Find the probability that the family has the following children. No girls

Step-by-Step Solution

Answer

The probability is \( \frac{1}{32} \).

Step 1: Understand the Probability Model

The situation involves a binomial probability model, where we have a fixed number of trials (5 in this case for 5 children), two possible outcomes (boy or girl), and the probability of one outcome (having a girl) is \( \frac{1}{2} \).

Step 2: Define the Event of Interest

We are specifically interested in the event where there are no girls in the family. This means all 5 children are boys. The probability of having a boy is also \( \frac{1}{2} \), since the probability of having a girl is \( \frac{1}{2} \).

Step 3: Calculate the Probability of Each Outcome

The probability of having one boy is \( \frac{1}{2} \). Thus, the probability of having 5 boys in a row is the product of the individual probabilities: \[ \left( \frac{1}{2} \right)^5 \]

Step 4: Compute the Final Probability

Calculate \( \left( \frac{1}{2} \right)^5 \) to find the probability that all 5 children are boys. \[ \left( \frac{1}{2} \right)^5 = \frac{1}{32} \]

Step 5: Interpret the Result

The probability that the family has 0 girls (all boys) is \( \frac{1}{32} \), which means this event is relatively rare.

Key Concepts

Probability Model

In a probability model, we aim to represent the situation using mathematical concepts to predict outcomes. Here, we're dealing with a binomial distribution model. This model involves a fixed number of trials, which in our case is 5, representing the number of children. Each child has two possible outcomes: being a boy or a girl. Since the probability of each gender is equal, each outcome has a probability of 0.5 or \( \frac{1}{2} \).

This binomial setting is perfect because each trial (or child) is independent of the others, and the probability remains constant for each trial. This model helps us calculate the likelihood of different occurrences, like having 0, 1, or all 5 girls. Understanding this model is essential, as it forms the backbone of determining the probabilities associated with these events.

Event Calculation

Event calculation involves determining the probability of a specific outcome based on the probability model. In the given problem, the event we are interested in is having no girls, meaning all 5 children are boys.

To calculate this probability, we need to consider the independent probability of each child being a boy, which is \( \frac{1}{2} \). Since we want all 5 children to be boys, we multiply the probability for each child together:

• The probability of 1 boy: \( \frac{1}{2} \)
• The probability of 5 boys: \( \left( \frac{1}{2} \right)^5 \)

Calculating \( \left( \frac{1}{2} \right)^5 = \frac{1}{32} \), we find that this represents the probability of having no girls. Event calculation simplifies complex situations into manageable numbers and helps quantify the likelihood of specific outcomes.

Interpretation of Probability

The interpretation of probability helps us understand what these numbers mean in real-world scenarios. In the exercise at hand, we calculated the probability of having no girls (all 5 children being boys) as \( \frac{1}{32} \).

This outcome, having a probability of \( \frac{1}{32} \), is relatively rare, indicating that such events do not often occur. When probabilities are expressed as fractions, smaller denominators suggest higher likelihoods, while larger ones (like 32) indicate less frequent occurrences.

Understanding probability interpretation is crucial because it allows you to assess how likely it is for specific events to happen, aiding decision-making processes and clarifying expectations in uncertain situations.