Problem 9 - Algebra 2

Question

School There are 8 girls and 8 boys on the Student Senate. Three of the students are seniors. What is the probability that a person selected from the student Senate is not a senior?

Step-by-Step Solution

Answer

The probability is \( \frac{13}{16} \).

Step 1: Define Total Number of Students

The Student Senate consists of 8 girls and 8 boys, making a total of 16 students.

Step 2: Identify Number of Seniors

According to the problem, there are 3 seniors on the Senate.

Step 3: Calculate Non-Seniors Total

Subtract the number of seniors from the total number of students to find the number of non-seniors: 16 (total students) - 3 (seniors) = 13 non-seniors.

Step 4: Calculate Probability of Selecting a Non-Senior

The probability of selecting a non-senior is calculated by dividing the number of non-seniors by the total number of students: \[ \frac{13}{16} \].

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with the arrangement, combination, and selection of objects. It is often used to understand probability by identifying all possible outcomes or configurations of a given scenario. In our exercise, combinatorics helps us determine the different groups of students—seniors and non-seniors—from the total number of students in the Student Senate.
You might think of combinatorics as organizing objects into specific orders or selections, which is crucial for calculating probabilities. For our purposes, since we know the total number of students and the number of seniors, we use simple subtraction to find the number of non-seniors. This approach streamlines the process of determining probabilities.

Student Senate

The Student Senate is essentially a representative body, similar to a governing council, comprising student members from a school. In our problem, the Senate includes both boys and girls, adding up to a total of 16 members. Understanding the structure of such a council is vital because it helps us to set the total number from which we are making a probability calculation.
The composition of the Student Senate, including sub-groups like boys, girls, seniors, and non-seniors, is essential for framing probabilities. This understanding allows us to calculate probabilities accurately by identifying how many members fall under specific categories, such as seniors or non-seniors.

Seniors

Seniors generally refer to students in their final year of school. They form a distinct subgroup within the Student Senate. In this problem, there are specifically 3 seniors among the 16 Student Senate members. Knowing the number of seniors allows us to determine the rest of the group's composition more accurately.
By isolating the seniors as a separate category, it becomes significantly easier to calculate the probability of selecting a student that does not fall into this group. It establishes a contrast necessary for calculating the likelihood of alternatives, which is crucial in probability questions.

Non-seniors

Non-seniors are students who are not in their final year of school. In the context of this exercise, identifying non-seniors is crucial for solving the probability question posed. After confirming that there are 3 seniors, we simply subtract this number from the total 16, revealing that there are 13 non-seniors in the Student Senate.
Once you know how many non-seniors are there, you can directly use this number in the probability formula: \[\text{Probability of selecting a non-senior} = \frac{\text{Number of non-seniors}}{\text{Total number of students}} = \frac{13}{16}\]
This shows the likelihood of picking a non-senior from the Student Senate, based on straightforward subtraction and division.