A professor thinks students who live on campus are more likely to get A in the probabilitycourse. To check this theory, the professor combines the data from the past few years:A. 600 students have taken the course.B. 120 students have got A.C. 200 students lived on campus.D. 80 students lived off campus and got A.Does this data suggest that “getting an A” and “living on campus” are dependent orindependent
Question
A professor thinks students who live on campus are more likely to get A in the probabilitycourse. To check this theory, the professor combines the data from the past few years:A. 600 students have taken the course.B. 120 students have got A.C. 200 students lived on campus.D. 80 students lived off campus and got A.Does this data suggest that “getting an A” and “living on campus” are dependent orindependent
Solution
To determine whether "getting an A" and "living on campus" are dependent or independent, we need to check if the probability of getting an A is affected by the event of living on campus.
From the data given:
- Total students = 600
- Students who got A = 120
- Students who lived on campus = 200
- Students who lived off campus and got A = 80
First, let's find the probability of a student getting an A, regardless of where they live. This is simply the number of students who got an A divided by the total number of students.
P(A) = Students who got A / Total students = 120 / 600 = 0.2
Next, let's find the probability of a student living on campus. This is the number of students who lived on campus divided by the total number of students.
P(Campus) = Students who lived on campus / Total students = 200 / 600 = 0.33
Now, let's find the probability of a student getting an A given that they live on campus. This is the number of students who lived on campus and got an A divided by the total number of students who lived on campus.
First, we need to find the number of students who lived on campus and got an A. We know that 80 students lived off campus and got an A, so the number of students who lived on campus and got an A is the total number of students who got an A minus the students who lived off campus and got an A.
Students who lived on campus and got A = Students who got A - Students who lived off campus and got A = 120 - 80 = 40
So, P(A|Campus) = Students who lived on campus and got A / Students who lived on campus = 40 / 200 = 0.2
If "getting an A" and "living on campus" were independent, then we would expect P(A|Campus) to be the same as P(A). Since P(A|Campus) = P(A), we can conclude that "getting an A" and "living on campus" are independent events. The data does not suggest that living on campus increases a student's chances of getting an A in the course.
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