Problem 48 - 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. 18 to 21 years old

Step-by-Step Solution

Answer

The probability that a randomly chosen student is between 18 to 21 years old is \( \frac{29}{32} \)

Step 1: Find the Total Number of Students in the Class

To find the total number of students, simply add the number of students in each age group. Thus, \(11 (from ages 18-19) + 18 (from ages 20-21) + 2 (from ages 22-30) + 1 (from ages 31-40) = 32\) students in total.

Step 2: Number of Students in the Age Group of 18-21

Count the number of students in the age group of 18-21. Here, two groups fall, 18-19 years old and 20-21 years old. By adding these together, \(11 (from ages 18-19) + 18 (from ages 20-21) = 29\) students.

Step 3: Calculate the Probability

The probability is the ratio of the favorable events to the total number of events. Therefore, in our scenario, the probability \(P\) that a student selected at random is between 18 and 21 years old is calculated as follows: \(P = \frac{29 (number of students in age group 18-21)}{32 (total number of students)} \).

Key Concepts

Age Group

Age groups help categorize individuals based on their age range. In this exercise, different age groups within the sociology class provide a framework to analyze and interpret probability data. Age groups in this example are:

• 18-19 years old
• 20-21 years old
• 22-30 years old
• 31-40 years old

Each group represents a distinct segment of the class population, making it easier to analyze the demographics. Understanding how age groups are distributed helps us make informed decisions or predictions about a class, like determining the probability of selecting a student from a specific age group.

Number of Students

The total number of students in each age group is crucial to determining probabilities. In this exercise, the steps to solve the probability rely on understanding each age group's population.

For example, we look at the:

• 11 students in the 18-19 age group
• 18 students in the 20-21 age group
• 2 students in the 22-30 age group
• 1 student in the 31-40 age group

By adding the number of students across all age groups, we derive the total number of students in the class, which is 32. Knowing this helps us compute probabilities, such as finding the likelihood of picking a student aged between 18 to 21.

Sociology Class

A sociology class serves as a microcosm of societal structures, including demographics examined through age groups. Analyzing a sociology class involves examining its composition – the group of students with varied backgrounds and ages.

Understanding the class dynamics can give insight into:

• The diversity of opinions or perspectives within the class
• The commonalities and differences in sociocultural experiences
• The probability of selecting students representing specific demographic segments

In essence, each class can reflect broader societal patterns, making statistical exercises like determining probabilities relevant in understanding real-world dynamics.