How do I know when to use addition or subtraction when doing compound event probability?

Addition Rule 1:

When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event.

P (A or B) = P (A) + P (B)

Addition Rule 2:

When two events, A and B, are non-mutually exclusive, the probability that A or B will occur is:

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

In the rule above, P (A and B) refers to the overlap of the two events.

A student goes to the library. The probability that she checks out (a) a work of fiction is 0.40, (b) a work of non-fiction is 0.30, and (c) both fiction and non-fiction is 0.20. What is the probability that the student checks out a work of fiction, non-fiction, or both?

Solution: Let F = the event that the student checks out fiction; and let N = the event that the student checks out non-fiction. Then, based on the rule of addition:

P (F ∪ N) = P (F) + P (N) - P (F ∩ N)

P (F ∪ N) = 0.40 + 0.30 - 0.20 = 0.50

Subtraction Rule:

The probability that event A will occur is equal to 1 minus the probability that event A will not occur.

P (A) = 1 - P (A')

Suppose, for example, the probability that Bill will graduate from college is 0.80. What is the probability that Bill will not graduate from college? Based on the rule of subtraction, the probability that Bill will not graduate is 1.00 - 0.80 or 0.20.