SOLUTION: In a class of 160 students, 90 are taking math, 78 are taking science, and 62 are taking both math and science. What is the probability of randomly choosing a student who is not ta

All the math students are in the red circle, labeled M for math.
All the science students are in the blue circle, labeled S for science.

Some are in both circles, and some are in neither circle.

The letter a represents the students who are in the red circle but not
in the blue circle. So a = the number of students taking math but not
taking science.

The letter c represents the students who are in the blue circle but not
in the red circle. So c = the number of students taking science but not
taking math.

The letter b represents the students who are in both the red circle and
also in the red circle. So b = the number of students taking both math
and science.

The letter d represents the students who are not in either circle. So b=
the number of students not taking math or science.

We look at the clue that is the most conclusive first,

So we knoe that b=62, so we write b=62 and put that in the middle
region:



Next we look at the clue that reads:

That means that there are 90 students altogether in the red circle. We already
have b=62 in the right part of the red circle, so there must be 90-62 or 28
in the left part of the red circle, so a=28, and we write that in the left
part of the red circle:




Next we look at the clue that reads:

That means that there are 78 students altogether in the blue circle. We already
have b=62 in the left part of the blue circle, so there must be 78-62 or 16
in the right part of the blue circle, so c=16, and we write that in the right
part of the blue circle:




Next we look at the remaining clue that reads:

We have accounted for a=28, b=62, and c=16 or 28+62+16 or 106 of the
160, so there are 160-106 or 54 remaining students who are not in 
either of the two circles, so d=54.  So we write d=54 in the region
outside of both circles:



The question is:

These are the students in either a=28 or d=54.  That's 28+54 or 82 students
that are not in the blue circle S.

So the probability is 82 out of 160 or  which reduces to .

Edwin