Question

WINK, Inc., conducted a demographic study of thousands of families that have exactly three biological children. What is the percentage probability that all the children in a randomly selected family will be the same gender? A. \(40 \%\) B. \(25 \%\) C. \(12.5 \%\) D. \(10 \%\)

Short answer

B. 25%

Step-by-step solution

Determine Possible Combinations

Calculate all possible gender combinations of three children. The combinations are: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG. Thus, there are 2 possible outcomes where all children are of the same gender (BBB and GGG).

Calculate Total Outcomes

Count the total possible combinations of three children. Since each child could be either a boy (B) or a girl (G), there are a total of 2^3 or 8 possible outcomes.

Determine Probability of Same Gender

Divide the number of favorable outcomes (all children same gender) by the total number of possible outcomes. There are 2 favorable outcomes (BBB and GGG) out of 8 total combinations. Therefore, the probability is \(\frac{2}{8} = \frac{1}{4} = 25\text{ percent}\).

Key concepts

Gender Combinations

Understanding gender combinations is the first step in solving problems related to the probability of family demographics. In a family with three children, each child can be either a boy (B) or a girl (G). This gives us a set of all possible combinations as follows:

• BBB
• BBG
• BGB
• BGG
• GBB
• GBG
• GGB
• GGG

There are eight possible combinations in total. This list helps us understand and count all the potential gender arrangements within a family of three children.

Probability Calculation

Probability calculation is a fundamental concept in statistics that allows us to determine the likelihood of various outcomes. In the context of gender combinations in a three-child family, we are interested in determining the probability that all three children are of the same gender.
To calculate this, we need to find the number of favorable outcomes (all children of the same gender) and divide it by the total number of possible outcomes. We have 2 favorable outcomes (BBB and GGG) and 8 possible outcomes in total. Therefore, the probability that all children are the same gender is:
\[\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{2}{8} = \frac{1}{4} = 25\%\]

Basic Combinatorics

Basic combinatorics is the study of counting, arranging, and combining objects according to specified rules. It's crucial for solving problems involving probabilities and other mathematical contexts. In our family gender combination problem:
We are using combinatorial methods to count possible sequences of genders for three children. Each child can be one of two genders (B or G), leading to a total of 2^3, or 8, combinations when considering three children.
These methods are foundational to many areas of mathematics and are particularly useful for calculating probabilities, as they help enumerate all possible outcomes systematically.

Demographic Study

A demographic study examines the structure of human populations using data such as birth rates, marriage rates, and other pertinent societal data. In the problem provided, WINK, Inc., conducted a demographic study on families with exactly three biological children. These studies are essential for understanding trends and patterns within populations.
From the data, we can calculate the probability of specific family compositions (such as the probability of all children being the same gender). These insights are valuable in several fields, including sociology, healthcare, and planning of social services.
Understanding probabilities in demographic studies helps policymakers and researchers predict future trends and make informed decisions.