Question

Arlene is testing whether school is more enjoyable when students are making high grades. She asked 100 students if they enjoyed school and whether their GPA was above or below 3.0. She found that 30 of the 40 students with a GPA above 3.0 reported that they enjoyed school, and 15 of the 60 students with a GPA below 3.0 reported that they enjoyed school. What is the probability that a student with a GPA below 3.0 does not enjoy school?

Answer 1

Well there's 100 students asked in total

40 students had a gpa above 3.0

60 had a gpa below 3.0

The question is asking for the ones who's gpa is below 3.0.
So the fraction of the students that don't enjoy the school would be

45/60

Answer 2

To find the probability that a student with a GPA below 3.0 does not enjoy school, we need to use the information given in the problem to calculate the number of students who have a GPA below 3.0 and do not enjoy school, and divide that by the total number of students with a GPA below 3.0.

From the problem statement, we know that:
- Arlene asked 100 students if they enjoyed school and whether their GPA was above or below 3.0.
- 30 of the 40 students with a GPA above 3.0 reported that they enjoyed school.
- 15 of the 60 students with a GPA below 3.0 reported that they enjoyed school.

Let's use the following variables to help us solve the problem:
- Let A be the event that a student enjoys school.
- Let B be the event that a student has a GPA below 3.0.

Using this notation, we know that:

P(A | GPA > 3.0) = 30/40 = 0.75
P(A | GPA < 3.0) = 15/60 = 0.25

We want to find P(not A | GPA < 3.0), which is the probability that a student with a GPA below 3.0 does not enjoy school. We can use Bayes' theorem to calculate this probability:

P(not A | GPA < 3.0) = P(GPA < 3.0 | not A) * P(not A) / P(GPA < 3.0)

To calculate the terms on the right-hand side of this equation, we need to use the following formulas:

P(not A) = 1 - P(A) = 1 - 0.25 = 0.75
P(GPA < 3.0) = P(GPA < 3.0 | A) * P(A) + P(GPA < 3.0 | not A) * P(not A)

We can calculate P(GPA < 3.0 | A) using the complement rule:

P(GPA < 3.0 | A) = 1 - P(GPA > 3.0 | A) = 1 - 0.75 = 0.25

Substituting these values into the formula for P(GPA < 3.0), we get:

P(GPA < 3.0) = 0.25 * 0.25 + P(GPA < 3.0 | not A) * 0.75

To solve for P(GPA < 3.0 | not A), we can rearrange this equation:

P(GPA < 3.0 | not A) = (P(GPA < 3.0) - 0.25 * 0.25) / 0.75
P(GPA < 3.0 | not A) = (0.25 - 0.0625) / 0.75
P(GPA < 3.0 | not A) = 0.25

Substituting this value and the other known values into the formula for P(not A | GPA < 3.0), we get:

P(not A | GPA < 3.0) = 0.25 * 0.75 / P(GPA < 3.0)
P(not A | GPA < 3.0) = 0.25 * 0.75 / (0.25 * 0.25 + 0.25 * 0.75)
P(not A | GPA < 3.0) = 0.75

Therefore, the probability that a student with a GPA below 3.0 does not enjoy school is 0.75.