Problem 34 - 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 boys

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to calculate the probability that all 5 children in a family are girls. The probability of each child being a girl is \( \frac{1}{2} \).

Step 2: Set Up the Probability Expression

Since each child is independent, the probability of having 5 girls is the product of the probabilities for each child being a girl. Thus, this is represented as \( \left( \frac{1}{2} \right)^5 \).

Step 3: Calculate the Probability

Calculate \( \left( \frac{1}{2} \right)^5 \), which means multiplying \( \frac{1}{2} \) by itself 5 times: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{32} \). This results in \( \frac{1}{32} \).

Key Concepts

Independent Events

When learning about probability, one key concept is independent events. These are events where the outcome of one does not affect the outcome of another. For example, imagine flipping a coin. Whether it lands on heads or tails does not influence what will happen on the next flip. Each flip is independent.

In the context of a family having children, if each child's gender is not influenced by their siblings, then each child's gender is an independent event. This is similar to the idea of flipping a coin, where each flip has an equal chance of resulting in heads or tails.

• The probability of having a girl is \( \frac{1}{2} \).
• The probability of having a boy is \( \frac{1}{2} \).

Understanding independent events helps simplify complex probability problems by allowing us to calculate the probability of multiple events occurring by simply multiplying their individual probabilities.

Probability of Multiple Events

In probability, when dealing with multiple independent events, you can find the probability of all of them happening by multiplying their individual probabilities. This is because each event occurs without influencing the others.

For instance, if you want to know the chance of flipping a coin and getting heads twice in a row:

• The probability of heads on the first flip: \( \frac{1}{2} \)
• The probability of heads on the second flip: \( \frac{1}{2} \)
• Probability of heads on both flips: \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)

In the scenario where a family desires all five children to be girls, the concept is applied with the probability equation \( \left( \frac{1}{2} \right)^5 \), calculating the chance of each child being a girl. This result, \( \frac{1}{32} \), shows how multiplying probabilities provides the overall likelihood of multiple independent events occurring together.

Binomial Probability

Binomial probability is a statistical method used to calculate the likelihood of a particular outcome, where there are two possible results for each trial. The trials are independent, and the number of trials is set, which suits our family-child scenario well.

When a family wants to predict the number of girls out of several children, each child can be viewed as an individual trial, with two possible outcomes: girl or boy. The probability remains constant for each child. The binomial formula for finding exactly "k" successes (or particular outcomes) in "n" trials is:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Where:

• \( P(X = k) \) is the probability of "k" successes.
• \( \binom{n}{k} \) is the binomial coefficient.
• "p" is the probability of success on any given trial.
• "n" is the number of trials.

In the family problem, if "n" is 5 and "p" is \( \frac{1}{2} \), you can use this formula to determine the likelihood for varying numbers of girls or boys, including the scenario of all children being girls, which we've calculated as \( \frac{1}{32} \) using basic probability rules for independent events.