In a class of 28 students, 15 earned an A on the final exam, 10 earned an A for the entire course, and 7 earned an A for both the final exam and the entire course. What is the probability that a randomly selected student in this class did not earn an A on the final exam and didn’t not earn an A for the entire course

We can solve this problem using set theory and probabilities.

### Step 1: Define Events
- Let \( A_1 \) represent the event that a student earned an A on the final exam.
- Let \( A_2 \) represent the event that a student earned an A for the entire course.
- From the problem:
  - \( |A_1| = 15 \) (students who earned an A on the final exam)
  - \( |A_2| = 10 \) (students who earned an A for the entire course)
  - \( |A_1 \cap A_2| = 7 \) (students who earned an A on both the final exam and the entire course)
  
The total number of students in the class is 28.

### Step 2: Use Inclusion-Exclusion to Find \( |A_1 \cup A_2| \)
To find the number of students who earned an A either on the final exam or for the entire course (or both), we use the inclusion-exclusion principle:
\[
|A_1 \cup A_2| = |A_1| + |A_2| - |A_1 \cap A_2|
\]
Substitute the known values:
\[
|A_1 \cup A_2| = 15 + 10 - 7 = 18
\]
So, 18 students earned an A on the final exam, for the course, or both.

### Step 3: Find the Complement (Neither A on the Exam nor the Course)
The number of students who did **not** earn an A on the final exam and did **not** earn an A for the entire course is the complement of \( A_1 \cup A_2 \):
\[
\text{Students who earned neither} = 28 - |A_1 \cup A_2| = 28 - 18 = 10
\

In a class of 28 students, 15 earned an A on the final exam, 10 earned an A for the entire course, and 7 earned an A for both the final exam and the entire course. What is the probability that a randomly selected student in this class did not earn an A on the final exam and didn’t not earn an A for the entire course?

Learn how to solve a probability problem using set theory and the inclusion-exclusion principle. In a class of 28 students, we calculate the number of students who did not earn an A on either the final exam or the course. Suitable for Grades 9-12.

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