SOLUTION: If 20 students register for a Sociology class and 35 students register for Psychology, and 15 students register for both classes, if one student is selected at random, what is the

 
Draw a circle which will enclose all 20 Sociology students. Label
it S.




Draw another circle, overlapping the first, which will enclose
all 35 Psychology students. Label it P.





The region where the two circle overlap will enclose the 15 students who
registered for both classes. So we begin by writing 15 in the overlapping
part to indicate that these 15 students are inside both circles.



Since the circle labeled S must contain 20 students, and 15 of these 
20 students are in the overlappng part, the other 20-15 or 5 students
in circle S must be in the left portion of circle S.  So we write 5
in the left portion of the left circle to indicate that the 5 students
are located there.



Those 5 and 15 students in the left circle account for the entire 20
Sociology students, the 5 taking Sociology and NOT Psychology, and
the 15 taking both courses.

Now, since the circle labeled P must contain 35 students, and 15 of these 
35 students are in the overlappng part, the other 35-15 or 20 students
in circle P must be in the right portion of circle P.  So we write 20
in the right portion of the right circle to indicate that the 20 students
are located there.


 
Now we see that we have a total of 5+15+20 or 40 students. 

We are asked for the probability that a randomly selected student
only registered for Sociology or Psychology.  That is, that they
only registered for ONE of the two courses.

So the random student selected must be from the left or right
regions and not from the middle region.

Since there are 5 students in the leftmost portion and 20 in the
rightmost region, to be succesful, we must pick one of these 5+20
or 25 out of the total 40.  So the desired probability is 
which reduces to .

Edwin