Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the probability that a family with three children will have the given composition. No girls

Step-by-Step Solution

Answer

The probability that a family with three children will have no girls is \(\frac{1}{8}\) or 12.5%.

Step 1: Identifying the possible gender combinations

In a family with three children, there are a total of eight possible combinations of genders (ignoring any other factors like twins or external factors that might affect gender probability). These are: 1. Boy, Boy, Boy (BBB) 2. Boy, Boy, Girl (BBG) 3. Boy, Girl, Boy (BGB) 4. Boy, Girl, Girl (BGG) 5. Girl, Boy, Boy (GBB) 6. Girl, Boy, Girl (GBG) 7. Girl, Girl, Boy (GGB) 8. Girl, Girl, Girl (GGG) In this problem, we want to find the probability of the family having no girls, which means we're looking for the probability of the first combination: Boy, Boy, Boy (BBB).

Step 2: Calculate the probability of a single event

Since there are two equally likely possibilities for each child's gender, we can assign a probability of 1/2 for each child being a boy and 1/2 for each child being a girl.

Step 3: Calculate the probability of having no girls

To find the probability of the family having no girls (BBB), we need to multiply the probabilities for each child being a boy independently. P(BBB) = P(Boy) × P(Boy) × P(Boy) = \(\frac{1}{2}\) × \(\frac{1}{2}\) × \(\frac{1}{2}\)

Step 4: Simplify the result

Now we just need to simplify the product: P(BBB) = \(\frac{1}{2}\) × \(\frac{1}{2}\) × \(\frac{1}{2}\) = \(\frac{1}{8}\) So, the probability that a family with three children will have no girls is 1/8 or 12.5%.

Key Concepts

Gender Combinations

In probability exercises involving multiple outcomes, it's important to identify all possible combinations first. When it comes to a family with three children, each child can be either a boy or a girl. This creates a tree diagram of outcomes, listing all combinations:

• Boy, Boy, Boy (BBB)
• Boy, Boy, Girl (BBG)
• Boy, Girl, Boy (BGB)
• Boy, Girl, Girl (BGG)
• Girl, Boy, Boy (GBB)
• Girl, Boy, Girl (GBG)
• Girl, Girl, Boy (GGB)
• Girl, Girl, Girl (GGG)

In total, there are 8 possible gender combinations. Each is equally likely if the chance of having a boy or girl is the same, usually set at 50/50 for simplicity. Understanding all possible outcomes helps us calculate the probability of each individual event.

Probability Calculation

Probability is about measuring how likely an event is to happen. For each child in the family, the probability of being a boy (\( P(Boy) \)) is \( \frac{1}{2} \) and the same for a girl (\( P(Girl) \)). To find the probability of a specific combination, like having no girls (BBB), you multiply the probability for each event together.

This means:

• Step 1: Probability of first child being a boy = \( \frac{1}{2} \)
• Step 2: Probability of second child being a boy = \( \frac{1}{2} \)
• Step 3: Probability of third child being a boy = \( \frac{1}{2} \)

Thus, \( P(BBB) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \).
Probability helps translate every gender combination into a quantifiable expectation.

Mathematical Simplification

Simplification is a key step in solving probability problems to make the answers more understandable. After calculating the combination probability, it's common to need simplification.

In our example, the calculated probability for having no girls (BBB) is \( \frac{1}{8} \).
Since the multiplication of fractions is straightforward:

• Multiply numerators: \( 1 \times 1 \times 1 = 1 \).
• Multiply denominators: \( 2 \times 2 \times 2 = 8 \).

Simplifying probabilities often provides insights, like seeing the result of \( \frac{1}{8} \) expresses the situation concisely, and even translates into a percentage: 12.5%. Understanding how to simplify keeps mathematical expressions clear and reveals the intrinsic likelihood of an event.