Please explain the steps for the following probability problem: among 500 freshmen pursuing a business degree at a university, 301 are enrolled in an economics course, 220 are enrolled in a mathematics course, and 130 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is neither enrolled in an economics course nor a mathematics course?

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We are given a population of 500 students. Of these, 301 take an economics course, 220 take a mathematics course, and 130 take both an economics and a mathematics course. We are asked to find the probability that a randomly chosen individual from this population is not enrolled in either economics or mathematics.

I. One approach is to use addition rule for non-mutually exclusive events.

`P(A " or " B)=P(A)+P(B)-P(A" and "B)`

If we let A= is enrolled in economics and B= is enrolled in mathematics then we have `P(A" or "B)=301/500 + 220/500 - 130/500 = 391/500`

This is the complement of the event "is not in either class."

So we want `1-391/500=109/500`

II. Another approach is to use a Venn diagram. The rectangle has 500 people in it. One of the overlapping circles has 301 people in it, 130 of which are in the overlap of the two circles. The other circle has 220 people in it, 130 of which are in the overlap.

Then we have 171+130+90=391 people inside the circles and 109 outside the circles. (See attachment.)

III Between the two classes we have accounted for 301+220=521 students. But we counted 130 students twice, so we actually have 521-130=391 students. Thus the remainder, 500-391=109, students are in neither class.

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