Question: A sample of 61 students was selected. Find the complements of the following events. Part 1 of 4…

A sample of 61 students was selected. Find the complements of the following events.

Part 1 of 4

Exactly 12 of the students have GPA higher than 2.7.

The complement is:

The number of students with a GPA higher than 2.7 is not equal to 12.

Part 2 of 4

At least 12 of the students have GPA higher than 2.7.

The complement is:

At most 11 students have GPA higher than 2.7.

Part 3 of 4

More than 12 of the students have GPA higher than 2.7.

The complement is:

Fewer than 13 students have GPA higher than 2.7.

Part 4 of 4

Fewer than 12 of the students have GPA higher than 2.7.

The complement is:

More than 11 students have GPA higher than 2.7.

Solution

Step 1 Let \( N = 61 \) be the total number of students. \[ A_1: \text{Number of students with GPA} > 2.7 = 12 \] \[ A_1^{c}: \text{Number of students with GPA} > 2.7 \neq 12 \] Step 2 \[ A_2: \text{Number of students with GPA} > 2.7 \geq 12 \] \[ A_2^{c}: \text{Number of students with GPA} > 2.7 \leq 11 \] Step 3 \[ A_3: \text{Number of students with GPA} > 2.7 > 12 \] \[ A_3^{c}: \text{Number of students with GPA} > 2.7 < 13 \] Step 4 \[ A_4: \text{Number of students with GPA} > 2.7 < 12 \] \[ A_4^{c}: \text{Number of students with GPA} > 2.7 > 11 \]