A high school is offering 3 classes: math, physics, and chemistry. The classes are open to any of the 100 students in the school. There are 28 students in the math class, 26 in the physics class, and 16 in the chemistry class. There are 12 students who are in both math and physics, 4 who are in both math and chemistry, and 6 who are in both physics and chemistry. In addition, there are 2 students taking all 3 classes. If a student is chosen randomly, what is the probability that he or she is not in any of the classes?
First you subtract 2 from 6, 4, and 12, because the 2 people that are taking all three courses are part of the people that are taking at least 2 courses.
You now have 4 people taking only chemistry and physics, 2 people taking only chemistry and math, and 10 people taking only physics and math.
Now we want to find the number of people taking only one course.
We have 26 people taking physics but 14 of them are also taking another course. So 26 - 14 = 12 people taking only physics.
We have 16 people taking chemistry but 6 of them are also taking another course. So 16 - 6 = 10 people taking only chemistry.
We have 28 people taking math but 12 of them are also taking another course. So 28 - 12 = 16 people taking only Math.
We have 2 + 2 + 4 + 10 + 10 + 12 + 16 = 56 people taking courses.
100 - 56 = 44 people taking no courses.
44/100 = 11/25 probability that he or she is not in any courses.