Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the table, which shows the age groups of students in a college sociology class. $$\begin{array}{|c|c|} \hline \text { Age } & \text { Number of students } \\ \hline 18-19 & 11 \\ \hline 20-21 & 18 \\ \hline 22-30 & 2 \\ \hline 31-40 & 1 \\ \hline \end{array}$$ A student from the class is randomly chosen for a project. Find the probability that the student is the given age. Younger than 31 years old

Step-by-Step Solution

Answer

The probability that a randomly chosen student is younger than 31 years old is \(31/32\).

Step 1: Identify the Total Number of Students

First, add up all the numbers in the 'Number of students' column to find the total number of students in the class. This can be done by calculating \(11 + 18 + 2 + 1 = 32\).

Step 2: Identify the Number of Students Younger Than 31

Students younger than 31 years old fall into the category of 18-19 years, 20-21 years, and 22-30 years age groups. Add up the number of students in these categories which is \(11 + 18 + 2 = 31\).

Step 3: Calculate the Probability

The probability of an event is calculated by dividing the number of desired outcomes by the number of total outcomes. In this case, the desired outcome is that a student is younger than 31 years old, and the total outcome is any student. Therefore, the probability can be calculated as \(31/32\).

Key Concepts

Probability Calculation

Probability is essential in various fields, including sociology, and it provides a way to predict the likelihood of certain events. When dealing with probability calculation, the primary principle is to understand the ratio of the desired outcome to all possible outcomes. In the context of our exercise, the desired outcome is choosing a student younger than 31 years old from a sociology class.

To calculate the probability, you start by counting all the students in the class, referred to as the 'total number of outcomes'. Then, you identify the count of students that meet the criteria of being younger than 31, which are the 'desired outcomes'. By dividing the number of desired outcomes by the total number of outcomes, you get the probability fraction. In a simplified form, probability is often expressed as a percentage to make it more intuitive to understand. For instance, a probability of \(31/32\) is very close to 1, suggesting a very high likelihood for the event to occur.

Sociology Class Demographics

Understanding the demographics of a sociology class can provide insights into the social dynamics and representativeness of the course study. Sociology often deals with various human aspects, including age, which can influence perspectives and contributions to the class discussions.

In the given exercise, we observe the age groups of students, which is an essential demographic feature. Sociologists would analyze this data to determine the diversity of the class or the proportion of students from different age brackets. This demographic analysis can help tailor teaching methods, study materials, or even highlight biases in class composition. For example, a class mainly composed of younger students might have different discourse tendencies compared to a more age-diverse classroom.

Age Group Statistical Analysis

Statistical analysis of age groups in a sociology class helps in quantifying characteristics of the population within that class. By looking at the range and distribution of ages, it becomes possible to discern patterns such as the majority age group, outliers, or the average age of the students.

In our exercise, the age groups are divided into four categories. Statistical analysis can include calculating the mean, median, or mode of these age groups. For instance, modes can be determined by seeing which age category has the highest frequency—in our case, the 20-21 age group. The analysis might further investigate whether the class age distribution reflects broader societal or academic trends, influencing the context and content of the educational experience.